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#!/usr/bin/perl

...
while (<>) {
    print if /^>?(\w+\n)/ && $names{$1};
}

$ ./extract.pl file1 file2 >output

$ perl -ne'$/eq"\n"?$nm{$_}++:/^>?(\w+\n)/&&$nm{$1}&&print;$/="\n>"if 
+eof' file1 file2 >output

$ perl -ne'1..eof&&($/=">")?${$_}++:/.+\n/&${$&}&&print' file1 file2 >
+output