Where is the mistake in the code ?

In addition to the incorrect regex you have a scoping problem with your $a. This is hidden from you because of your choice of variable name. Never use $a or $b as a general variable name in perl - they are special variables used for sort routines. If you replace your $a with $num for example then your code won't compile:

$ cat scoping.pl 
#!/usr/bin/env perl 
use strict;
use warnings;


my $data = <<'EOD';

<stlib:mem_bit>72</stlib:mem_bit>
EOD

open my $fh, '<', \$data;

while (my $line = <$fh>) {
    

while ( $line =~ /.*st_mem_bit.*(\d+)/g ) {
    my $num= $1;
    

}
}
print $num;
close $fh;

$ ./scoping.pl 
Global symbol "$num" requires explicit package name at ./scoping.pl li
+ne 22.
Execution of ./scoping.pl aborted due to compilation errors.
$
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See Coping with scoping for why this is.

FWIW, I'm happy to add my voice to those suggesting that an XML parser is almost certainly a better approach for this task.