Yes, it does.

$ perl -le'"foo" =~ /(foo)/; print $1; "bar" =~ /(bar)/; print $1;'
foo
bar
[download]

What you got wrong is that a failed match will not change the values:

$ perl -le'"foo" =~ /(foo)/; print $1; "bar" =~ /(bar)/; print $1; "ba
+z" =~ /(quux)/; print $1;'
foo
bar
bar
[download]

Of course then, if all you're matching for is /(foo)/, and you succeed on the very first match, $1 will never be anything other than foo. What you need to do is check whether your match succeeded:

while(<DATA>) {
    if( $_ =~ /(foo)/ ) {
        print "$.: $1 -- $&\n";
    }
    else {
        print "No match on line $..\n"
    }
}
[download]

This is documented behaviour.

As a sidenote, instead of keeping a line counter yourself, you can use Perl's $. variable, which contains the number of the last line read from the last accessed filehandle.