I think you understand ^ and $ just fine; if /m is in effect, there is where they can match:

"AAC\nGTT\n"
 ^  $$^  $$
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But the regex only searches for the first match, and because the dot doesn't match the \n (it would only do that with /s), it goes from A to C. If you ask perl to do a second match, it will find GTT:

$ perl -wle 'print $& while "AAC\nGTT"=~/^.*$/mg;'
AAC
GTT
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If you want to match the second line straight away, you can do something like this:

$ perl -wle '"AAC\nGTT"=~/.*^(.*)$/ms; print $1'
GTT
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