It's probably not such a wonderful idea to change the input record separator in the middle of the loop, twice. You're confusing yourself doing that. Plus the changes are falling out of scope before the next iteration anyway.

Here's what you're reading into $_ on each iteration: First iteration: $_ = <a>a,\n. Then partway into the iteration, you change the input record separator, but it doesn't matter because you've already read your first line of data. Then a little further into the loop you change the input record separator again, and again it makes no difference since you already read the line of data. Then your localization of $/ falls out of scope, and the input record separator reverts back to "\n".

Now you read in b</a>\n. And so on.

You are still reading your data one line at a time so your pattern matches are matching against one line at a time.

To see what is being read, you might experiment with the following program:

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

while (<DATA>) {
    print "start of loop\n";
    print "\$_ = \"$_\"\n";
}

__DATA__
<a>a,
b</a>
<b>a,b</b>
<b>a,b
</b>
[download]

To make <DATA> read all your DATA instead of just one line, you can set $/ before your while loop, as follows. Note the use of a block ({ }) and local so that $/ isn't affected elsewhere in the program.

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

{
    local $/;
    while (<DATA>) {
        print "start of loop\n";
        print "\$_ = \"$_\"\n";
    }
}

__DATA__
<a>a,
b</a>
<b>a,b</b>
<b>a,b
</b>
[download]

With $/ set to undef (which is what local $/ does), there is no need for the while loop - all the data is read on the first iteration. You can read all the data into a single variable as follows:

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

my $var = do { local $/; <DATA> };
print "\$var = \"$var\"";

__DATA__
<a>a,
b</a>
<b>a,b</b>
<b>a,b
</b>
[download]

Then you can substitute against this variable as follows:

$var =~ s!<a>(.*?),(.*?)</a>!<a>$1A$2</a>!gs;
[download]

When you get that working, you might want to try with the following data:

__DATA__
This is to test.
<a>a,
b</a>
<b>a,b</b>
<b>a,b
</b>
<a>ab</a>
<c>a,b</c>
<a>a,b</a>
[download]

dear ig, thanks a lot for your great explanation.