One useful thing to think about with grep is what it will look like as a loop.

my @grepnames = grep {/"ER"/} @test2;
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Is the same as

my @grepnames;
for my $test2item (@test2) {
    push @grepnames,$test2item if $test2item =~ /"ER"/;
}
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After looking at it in this format, I believe it's easier to notice that we're trying to match each item against an arrayref instead of the first element in the array.

Another thing I noticed is I think you're confused on how the eq operator works. This operator sees if the string before the operator is exactly the same as the string after the operator. So the only thing that would evaluate to true in the comparison is "ER" eq "ER".

The last thing I noticed is that regexps treat quote characters as literal quote characters instead of designating a string. So "XERXES" wouldn't match the pattern /"ER"/ but 'X"ER"XES' would.