Take a look to eof
In a while (<>) loop, eof or eof(ARGV) can be used to detect the end of each file, whereas eof() will detect the end of the very last file only.

Update: try this not tested version
 perl -walne '$;="\t";$i||=0;$h{$i,$F[0]}.="$F[1]" } continue { $i++ if eof; END{print "$_:\t" . $h{$_} for sort keys %h}' file1.txt file2.txt

Seems OK

$ perl -MO=Deparse -walne '$;="\t";$i||=0;$h{$i,$F[0]}.="$F[1]" } cont
+inue { $i++ if eof; END{print "$_:\t" . $h{$_} for sort keys %h}' fil
+e1.txt file2.txt
BEGIN { $^W = 1; }
BEGIN { $/ = "\n"; $\ = "\n"; }
LINE: while (defined($_ = <ARGV>)) {
    chomp $_;
    our(@F) = split(' ', $_, 0);
    $; = "\t";
    $i ||= 0;
    $h{join $;, $i, $F[0]} .= "$F[1]";
}
continue {
    ++$i if eof;
}
sub END {
    print "${_}:\t" . $h{$_} foreach (sort keys %h);
}
-e syntax OK
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