In which folder should I install Composer on a shared linux server?
TL;DR My recommendation would be /usr/local/bin

It doesn't really matter as long as the directory is in the PATH of all users, and as long as it is not one of the locations which the OS wants full control over (those would be primarily /usr/bin and /bin).  For most situations, a reasonable location is /usr/local/bin because it should already be in the system's default PATH and it is intended precisely for this purpose -- local installations which are not part of the OS proper.

For Linux, you might want to refer to the LHS entry; this standard codifies usage inherited from the original Unix.

If it's convenient for you, installing in $HOME/bin and then moving to a system-wide location is fine.  In fact, most make install etc facilities expect you to be doing something like this.