I believe the only other such operators are + and ++. And that the parsing has worked this way forever, certainly on 1.0:
julia> :(1*2*3) |> dump
Expr
head: Symbol call
args: Array{Any}((4,))
1: Symbol *
2: Int64 1
3: Int64 2
4: Int64 3
julia> :(1+2+3)
:(1 + 2 + 3)
julia> :(1++2++3)
:(1 ++ 2 ++ 3)
julia> :(1/2/3)
:((1 / 2) / 3)
julia> VERSION
v"1.0.5"
For + and *, there are fallback definitions which work left-to-right, e.g. @less 1 * 2 * 3. What is new on 1.7 is that some chains of matrix multiplication will be done in other orders if this is more efficient. (Before this, there were also some other specialisations such as @which [1,2]' * Diagonal([3,4]) * [5,6].)