It is not completely clear what you wish to extract. This will extract the variable parts. It assumes a name value may not contain a double-quote. If that is not correct match on the quote and following tag, like the second half of the regex.

#!/usr/bin/perl
use strict;
use warnings;


$_ = q|<a name="a name"></a><!--start-->some
stuff<!--end-->| .
  q|<a name="Mae B Arthur"></a><!--start-->various text<!--end-->| .
  q|<a name="36561357542"></a><!--start-->What kind of #'s that<!--end
+-->| .
  q|<a name="aafq0w4tyu89[ "></a><!--start-->aeo;utrq[134[ a<!--end-->
+| ;



while ( m|<a name="([^"]+?)"></a>(?:<!--start-->(.*?)(<!--end-->)+?)|s
+gc ) {
    print "name: |$1|\n" .
          "art:  |$2|\n\n";
}
__DATA__
name: |a name|
art:  |some
stuff|

name: |Mae B Arthur|
art:  |various text|

name: |36561357542|
art:  |What kind of #'s that|

name: |aafq0w4tyu89[ |
art:  |aeo;utrq[134[ a|
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