What are "file1" and "file2", typeglobs, strings, barewords? Perl doesn't know, since you didn't give any indication. So it must guess.

It guessed an unquoted string. You then opened, bareword filehandle. What you've stored in @filez would probably suffice if you weren't using strictures, but with strictures on, you are informed that you're using a symbolic reference, which is considered illegal, and the script dies. (Updated)

You then print to the filehandle stored in $file. But $file holds a string, not a filehandle.

You probably want something like this:

my @files = \( *FILE1, *FILE2 );
open FILE1, ">$path1" or die $!;
open FILE2, ">$path2" or die $!;
foreach my $file ( @files ) {
    print $file "Some data\n";
}
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