If you examine the elements of $aa carefully, you'll find they all have trailing whitespace. This is a consequence of doing the split on "`n". If you trim them after you do the split, you'll get the expected result.
$a = @"
00013120747
00013051436
00013110491
00002100011
"@
$aa = $a.Split("`n") |% {$_.trim()}
$aa[2] -eq "00013110491"
True
The -match will match anywhere in the string, so it will still match even with the trailing space. The -eq requires and exact match, character for character, and the trailing space will cause it to return False.