Have you printed the two values you get?

The output is above in the end block.

I'm pretty sure that by "the two values", Corion meant $regex1 and $regex2.

By the way, the only quoting construct that doesn't interpolate at all is here docs with a single-quoted delimiter:

chomp( my $regex2 = <<'ENDREGEX' );
(?:(?|(?:\")([^\\\"]*(?:\\.[^\\\"]*)*)(?:\")))
ENDREGEX
print $regex2, "\n";

my ($match2) = q|"hubba \"bubba\""| =~ /$regex2/;
print $match2, "\n";

__END__

(?:(?|(?:\")([^\\\"]*(?:\\.[^\\\"]*)*)(?:\")))
hubba \"bubba\"
[download]

... string that behaves just as the compiled regex ...

That's what  qr// is for; see perlop. The  Regexp object returned by qr// is almost a string; it interpolates as a string in  m// s/// qr// expressions or when print-ed or string interpolated. It is not, however, the same as what is produced by the  qq// q// operators.