Just to show that I did not fat-finger this, or if there is any additional clue that can help debug the issue, here is a full capture
$ julia
               _
   _       _ _(_)_     |  Documentation: https://docs.julialang.org
  (_)     | (_) (_)    |
   _ _   _| |_  __ _   |  Type "?" for help, "]?" for Pkg help.
  | | | | | | |/ _` |  |
  | | |_| | | | (_| |  |  Version 1.9.2 (2023-07-05)
 _/ |\__'_|_|_|\__'_|  |
|__/                   |
julia> tempdir()
"/tmp/davide"
julia> import Pkg; Pkg.add("Revise")
  Installing known registries into `~/.julia`
┌ Warning: could not download https://pkg.julialang.org/registries
│   exception = RequestError: Recv failure: Connection reset by peer while requesting https://pkg.julialang.org/registries
└ @ Pkg.Registry /home/sw/spack-v0.20.1-b/opt/spack/linux-rhel8-icelake/gcc-13.1.0/julia-1.9.2-yux5s54oyok6xiifl5xwqeqjljazifjg/share/julia/stdlib/v1.9/Pkg/src/Registry/Registry.jl:69
┌ Warning: could not download https://pkg.julialang.org/registries
│   exception = RequestError: Recv failure: Connection reset by peer while requesting https://pkg.julialang.org/registries
└ @ Pkg.Registry /home/sw/spack-v0.20.1-b/opt/spack/linux-rhel8-icelake/gcc-13.1.0/julia-1.9.2-yux5s54oyok6xiifl5xwqeqjljazifjg/share/julia/stdlib/v1.9/Pkg/src/Registry/Registry.jl:69
     Cloning registry from "https://github.com/JuliaRegistries/General.git"
       Added registry `General` to `~/.julia/registries/General`
┌ Warning: could not download https://pkg.julialang.org/registries
│   exception = RequestError: Recv failure: Connection reset by peer while requesting https://pkg.julialang.org/registries
└ @ Pkg.Registry /home/sw/spack-v0.20.1-b/opt/spack/linux-rhel8-icelake/gcc-13.1.0/julia-1.9.2-yux5s54oyok6xiifl5xwqeqjljazifjg/share/julia/stdlib/v1.9/Pkg/src/Registry/Registry.jl:69
    Updating registry at `~/.julia/registries/General`
    Updating git-repo `https://github.com/JuliaRegistries/General.git`
   Resolving package versions...
┌ Warning: could not download https://pkg.julialang.org/registries
│   exception = RequestError: Recv failure: Connection reset by peer while requesting https://pkg.julialang.org/registries
└ @ Pkg.Registry /home/sw/spack-v0.20.1-b/opt/spack/linux-rhel8-icelake/gcc-13.1.0/julia-1.9.2-yux5s54oyok6xiifl5xwqeqjljazifjg/share/julia/stdlib/v1.9/Pkg/src/Registry/Registry.jl:69
   Installed Requires ─────────── v1.3.0
   Installed CodeTracking ─────── v1.3.5
   Installed LoweredCodeUtils ─── v2.3.0
   Installed JuliaInterpreter ─── v0.9.26
   Installed OrderedCollections ─ v1.6.2
   Installed Revise ───────────── v3.5.7
    Updating `~/.julia/environments/v1.9/Project.toml`
  [295af30f] + Revise v3.5.7
    Updating `~/.julia/environments/v1.9/Manifest.toml`
  [da1fd8a2] + CodeTracking v1.3.5
  [aa1ae85d] + JuliaInterpreter v0.9.26
  [6f1432cf] + LoweredCodeUtils v2.3.0
  [bac558e1] + OrderedCollections v1.6.2
  [ae029012] + Requires v1.3.0
  [295af30f] + Revise v3.5.7
  [0dad84c5] + ArgTools v1.1.1
  [56f22d72] + Artifacts
  [2a0f44e3] + Base64
  [ade2ca70] + Dates
  [8ba89e20] + Distributed
  [f43a241f] + Downloads v1.6.0
  [7b1f6079] + FileWatching
  [b77e0a4c] + InteractiveUtils
  [b27032c2] + LibCURL v0.6.3
  [76f85450] + LibGit2
  [8f399da3] + Libdl
  [56ddb016] + Logging
  [d6f4376e] + Markdown
  [ca575930] + NetworkOptions v1.2.0
  [44cfe95a] + Pkg v1.9.2
  [de0858da] + Printf
  [3fa0cd96] + REPL
  [9a3f8284] + Random
  [ea8e919c] + SHA v0.7.0
  [9e88b42a] + Serialization
  [6462fe0b] + Sockets
  [fa267f1f] + TOML v1.0.3
  [a4e569a6] + Tar v1.10.0
  [cf7118a7] + UUIDs
  [4ec0a83e] + Unicode
  [deac9b47] + LibCURL_jll v7.84.0+0
  [29816b5a] + LibSSH2_jll v1.10.2+0
  [c8ffd9c3] + MbedTLS_jll v2.28.2+0
  [14a3606d] + MozillaCACerts_jll v2022.10.11
  [83775a58] + Zlib_jll v1.2.13+0
  [8e850ede] + nghttp2_jll v1.48.0+0
  [3f19e933] + p7zip_jll v17.4.0+0
julia> ┌ Error: Pkg.precompile error
│   exception =
│    IOError: stat("/tmp/aaa/spack-stage/spack-stage-julia-1.9.2-yux5s54oyok6xiifl5xwqeqjljazifjg/spack-src/usr/share/julia/stdlib/v1.9/MbedTLS_jll/src/MbedTLS_jll.jl"): permission denied (EACCES)
│    Stacktrace:
│      [1] uv_error
│        @ ./libuv.jl:100 [inlined]
│      [2] stat(path::String)
│        @ Base.Filesystem ./stat.jl:152
│      [3] samefile
│        @ ./stat.jl:476 [inlined]
│      [4] stale_cachefile(modkey::Base.PkgId, build_id::UInt128, modpath::String, cachefile::String; ignore_loaded::Bool)
│        @ Base ./loading.jl:2818
│      [5] stale_cachefile
│        @ ./loading.jl:2717 [inlined]
│      [6] #stale_cachefile#954
│        @ ./loading.jl:2715 [inlined]
│      [7] stale_cachefile
│        @ ./loading.jl:2714 [inlined]
│      [8] _is_stale!(stale_cache::Dict{Tuple{Base.PkgId, UInt128, String, String}, Bool}, paths::Vector{String}, sourcepath::String)
│        @ Pkg.API /home/sw/spack-v0.20.1-b/opt/spack/linux-rhel8-icelake/gcc-13.1.0/julia-1.9.2-yux5s54oyok6xiifl5xwqeqjljazifjg/share/julia/stdlib/v1.9/Pkg/src/API.jl:1061
│      [9] macro expansion
│        @ /home/sw/spack-v0.20.1-b/opt/spack/linux-rhel8-icelake/gcc-13.1.0/julia-1.9.2-yux5s54oyok6xiifl5xwqeqjljazifjg/share/julia/stdlib/v1.9/Pkg/src/API.jl:1415 [inlined]
│     [10] (::Pkg.API.var"#245#276"{Bool, Bool, Pkg.Types.Context, Vector{Task}, Dict{Base.PkgId, String}, Dict{Base.PkgId, IOBuffer}, Pkg.API.var"#color_string#266", Base.Event, Base.Event, ReentrantLock,
Vector{Base.PkgId}, Vector{Base.PkgId}, Dict{Base.PkgId, String}, Vector{Base.PkgId}, Vector{Base.PkgId}, Dict{Base.PkgId, Bool}, Dict{Base.PkgId, Base.Event}, Dict{Base.PkgId, Bool}, Vector{Pkg.Types.Packa
geSpec}, Dict{Base.PkgId, String}, Dict{Tuple{Base.PkgId, UInt128, String, String}, Bool}, Vector{Base.PkgId}, Bool, Base.TTY, Base.Semaphore, String, Vector{String}, Vector{Base.PkgId}, Base.PkgId})()
│        @ Pkg.API ./task.jl:514
└ @ Pkg.API /home/sw/spack-v0.20.1-b/opt/spack/linux-rhel8-icelake/gcc-13.1.0/julia-1.9.2-yux5s54oyok6xiifl5xwqeqjljazifjg/share/julia/stdlib/v1.9/Pkg/src/API.jl:1302
┌ Error: Pkg.precompile error
│   exception =
│    IOError: stat("/tmp/aaa/spack-stage/spack-stage-julia-1.9.2-yux5s54oyok6xiifl5xwqeqjljazifjg/spack-src/usr/share/julia/stdlib/v1.9/Zlib_jll/src/Zlib_jll.jl"): permission denied (EACCES)
│    Stacktrace:
│      [1] uv_error
│        @ ./libuv.jl:100 [inlined]
│      [2] stat(path::String)
│        @ Base.Filesystem ./stat.jl:152
│      [3] samefile
│        @ ./stat.jl:476 [inlined]
│      [4] stale_cachefile(modkey::Base.PkgId, build_id::UInt128, modpath::String, cachefile::String; ignore_loaded::Bool)
│        @ Base ./loading.jl:2818
│      [5] stale_cachefile
│        @ ./loading.jl:2717 [inlined]
│      [6] #stale_cachefile#954
│        @ ./loading.jl:2715 [inlined]
│      [7] stale_cachefile
│        @ ./loading.jl:2714 [inlined]
│      [8] _is_stale!(stale_cache::Dict{Tuple{Base.PkgId, UInt128, String, String}, Bool}, paths::Vector{String}, sourcepath::String)
│        @ Pkg.API /home/sw/spack-v0.20.1-b/opt/spack/linux-rhel8-icelake/gcc-13.1.0/julia-1.9.2-yux5s54oyok6xiifl5xwqeqjljazifjg/share/julia/stdlib/v1.9/Pkg/src/API.jl:1061
│      [9] macro expansion
│        @ /home/sw/spack-v0.20.1-b/opt/spack/linux-rhel8-icelake/gcc-13.1.0/julia-1.9.2-yux5s54oyok6xiifl5xwqeqjljazifjg/share/julia/stdlib/v1.9/Pkg/src/API.jl:1415 [inlined]
│     [10] (::Pkg.API.var"#245#276"{Bool, Bool, Pkg.Types.Context, Vector{Task}, Dict{Base.PkgId, String}, Dict{Base.PkgId, IOBuffer}, Pkg.API.var"#color_string#266", Base.Event, Base.Event, ReentrantLock,
Vector{Base.PkgId}, Vector{Base.PkgId}, Dict{Base.PkgId, String}, Vector{Base.PkgId}, Vector{Base.PkgId}, Dict{Base.PkgId, Bool}, Dict{Base.PkgId, Base.Event}, Dict{Base.PkgId, Bool}, Vector{Pkg.Types.Packa
geSpec}, Dict{Base.PkgId, String}, Dict{Tuple{Base.PkgId, UInt128, String, String}, Bool}, Vector{Base.PkgId}, Bool, Base.TTY, Base.Semaphore, String, Vector{String}, Vector{Base.PkgId}, Base.PkgId})()
│        @ Pkg.API ./task.jl:514
└ @ Pkg.API /home/sw/spack-v0.20.1-b/opt/spack/linux-rhel8-icelake/gcc-13.1.0/julia-1.9.2-yux5s54oyok6xiifl5xwqeqjljazifjg/share/julia/stdlib/v1.9/Pkg/src/API.jl:1302
┌ Error: Pkg.precompile error
│   exception =
│    IOError: stat("/tmp/aaa/spack-stage/spack-stage-julia-1.9.2-yux5s54oyok6xiifl5xwqeqjljazifjg/spack-src/usr/share/julia/stdlib/v1.9/LibSSH2_jll/src/LibSSH2_jll.jl"): permission denied (EACCES)
┌ Error: Pkg.precompile error
│   exception =
│    IOError: stat("/tmp/aaa/spack-stage/spack-stage-julia-1.9.2-yux5s54oyok6xiifl5xwqeqjljazifjg/spack-src/usr/share/julia/stdlib/v1.9/LibSSH2_jll/src/LibSSH2_jll.jl"): permission denied (EACCES)
│    Stacktrace:
│      [1] uv_error
│        @ ./libuv.jl:100 [inlined]
│      [2] stat(path::String)
│        @ Base.Filesystem ./stat.jl:152
│      [3] samefile
│        @ ./stat.jl:476 [inlined]
│      [4] stale_cachefile(modkey::Base.PkgId, build_id::UInt128, modpath::String, cachefile::String; ignore_loaded::Bool)
│        @ Base ./loading.jl:2818
│      [5] stale_cachefile
│        @ ./loading.jl:2717 [inlined]
│      [6] #stale_cachefile#954
│        @ ./loading.jl:2715 [inlined]
│      [7] stale_cachefile
│        @ ./loading.jl:2714 [inlined]
│      [8] _is_stale!(stale_cache::Dict{Tuple{Base.PkgId, UInt128, String, String}, Bool}, paths::Vector{String}, sourcepath::String)
│        @ Pkg.API /home/sw/spack-v0.20.1-b/opt/spack/linux-rhel8-icelake/gcc-13.1.0/julia-1.9.2-yux5s54oyok6xiifl5xwqeqjljazifjg/share/julia/stdlib/v1.9/Pkg/src/API.jl:1061
│      [9] macro expansion
│        @ /home/sw/spack-v0.20.1-b/opt/spack/linux-rhel8-icelake/gcc-13.1.0/julia-1.9.2-yux5s54oyok6xiifl5xwqeqjljazifjg/share/julia/stdlib/v1.9/Pkg/src/API.jl:1415 [inlined]
│     [10] (::Pkg.API.var"#245#276"{Bool, Bool, Pkg.Types.Context, Vector{Task}, Dict{Base.PkgId, String}, Dict{Base.PkgId, IOBuffer}, Pkg.API.var"#color_string#266", Base.Event, Base.Event, ReentrantLock,
Vector{Base.PkgId}, Vector{Base.PkgId}, Dict{Base.PkgId, String}, Vector{Base.PkgId}, Vector{Base.PkgId}, Dict{Base.PkgId, Bool}, Dict{Base.PkgId, Base.Event}, Dict{Base.PkgId, Bool}, Vector{Pkg.Types.Packa
geSpec}, Dict{Base.PkgId, String}, Dict{Tuple{Base.PkgId, UInt128, String, String}, Bool}, Vector{Base.PkgId}, Bool, Base.TTY, Base.Semaphore, String, Vector{String}, Vector{Base.PkgId}, Base.PkgId})()
│        @ Pkg.API ./task.jl:514
└ @ Pkg.API /home/sw/spack-v0.20.1-b/opt/spack/linux-rhel8-icelake/gcc-13.1.0/julia-1.9.2-yux5s54oyok6xiifl5xwqeqjljazifjg/share/julia/stdlib/v1.9/Pkg/src/API.jl:1302
julia>
Note that sw is not an user of this machine and that /home/sw/ is only a location shared for everybody to read (as /opt is on other machines).
Note also (not shown) that the directory /tmp/davide exists and is writable by current user.
Thanks a lot to anybody who will provide additional insight!