What is a drawback of such a variable, say for multithreading,

If it is read-only. Ie. just conveying some piece of static information from the main program to the modules/threads, there is no significant problem.

If it is intended to act as a conduit for passing dynamic information back and forth between threads, then you would need to apply locking.

In general, there is no difference between obtaining a piece of static information from a global variable, and calling a (global) function (singleton instantiator) to obtain that information. Both involve the calling code "knowing" an well-known name.

Eg. $My::Config::DBhandle->somemethod( ... );

or My::Config->new()->get_DBHandle()->somemethod( ... );