My assumption is if the shebang is not mandatory for pm files in Linux (because pm files are not meant to be executed from command line), then I guess the file's permissions can be 644. They are not required to be 755.

Yes. For cheap thrills, I ran these commands on my recently installed perl 5.38 on Ubuntu:

~/localperl$ find . -name '*.pm' -ls >pm.tmp
~/localperl$ find . -name '*.pl' -ls >pl.tmp
[download]

All the .pm files have permissions like this:

-r--r--r--   ... ./lib/5.38.0/Carp.pm
[download]

The perl commands in the bin directory designed to be run as commands have permissions like this example of the prove command:

-rwxr-xr-x  ... prove
[download]

These text files also tend to start with a shebang line like:

#!/home/youruser/localperl/bin/perl
[download]

There are also plenty of .pl files lying around with permissions like this:

-r--r--r-- ... somefile.pl
[download]

which is fine because you can still run them manually via:

perl /somepath/somefile.pl
[download]