When I got deep recursion, I tried changing my regex, but I see that I just made it match the entire string. Is the problem still with my regex or am I using recursion incorrectly?

I don't see any need for recursion at all. As far as I can see, everything you want to do can be done in a properly constituted regex. There may be speed considerations that make a different approach preferable, but they can only be addressed by careful specification and benchmarking.

Update: To directly address your question about recursion:

sub find_coding {
         my ( $seq ) = @_;
         while ( $seq =~ /($regex)/g ) {
                 my $match = $1;
                 my $sequence = find_coding($match);
                 my $allmatch = find_coding($sequence);
                 print "Coding region: $allmatch", "\n";
         }
 }
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When you find any match of  $seq against the regex, you take the matching subsequence  $1 and feed it back to the function again:
    my $match = $1;
    my $sequence = find_coding($match);
Of course, this subsequence matches again because it already matched, and we're off to infinity... and beyond!

Note also that the
    my $sequence = find_coding($match);
statement and the following
    my $allmatch = find_coding($sequence);
statement (were it ever to be executed) are meaningless because the  find_coding() function doesn't return anything meaningful; as best I can see, it would return (if it ever did) the return value of the print statement at the end of the while-loop. (Update: Changed this paragraph to address both statements.)

Update:

...the  find_coding() function ... would return ... the return value of the print statement at the end of the while-loop.

On second thought, I take this back. Without an explicit return statement, a function returns the value of the last expression executed in the function. My idea was that the print statement at the end of the while-loop would be that last expression. In fact, if the function didn't infinitely recurse, the last expression executed in the function would be the  $seq =~ /($regex)/g regex in the conditional of the while-loop: when this expression evaluates false, the loop will exit and, as there is no code in the function after the loop, the function will exit and return this ever-false value.