It looks like an ugly bug.
Herein a workaround for loglog plots with aspect ratio = 1.
NB: edited code as there seems to be another bug, with aspect_ratio not being respected in loglog plots if range of xlims != ylims
using Plots; gr(legend=false)
# Define some input data:
x, y = 0.1, 0.1 # vector origin
u, v = 9.9, 99.9 # vector
xl, yl = (0.01, 100), (0.01, 200) # loglog plot limits (have to be modified to max range)
# plot arrowheads as rotated unicode characters (OK for ratio=1):
lminmax = (minimum([xl[1];yl[1]]), maximum([xl[2];yl[2]]))
r = atan(log10.(1+v/y), log10.(1+u/x)) * 180/pi # rotation angle
p = plot([x,x+u], [y,y+v], lc=:blue, xlabel="X",ylabel="Y")
plot!(ratio=1, scale=:log10, xlims=lminmax, ylims=lminmax)
annotate!(x+u,y+v, text('\u27A4', 9, :blue, rotation=r))
annotate!(x,y, text('\u27A4', 9, :blue, rotation=r+180))
NB: it should be possible to correct the rotation angles for the general case where ratio != 1