The only minor modification, I would do - keep the header line

That would not process in the code below - it is saving genes and their values - if you want to save the headers the code would need to be rewritten to accomplish that.

while (<>) {
    next if /^ID/;
    chomp;
    
    # if $ARGV = 'tab1.txt', $table will = 'tab1'
    # unless $table is already initiallized.
    # $table is uninitiallized on the first read
    # and after end of file for every file
    # (see below: $table = '' if eof;)
    ($table = fileparse $ARGV) =~ s/\.txt$// unless $table;
    
    # If the column count for this table is not already
    # stored, then use the transliteration operator to count
    # the number of pipes (# of columns) and store it.
    $col_cnt{$table} ||=  tr/|//;
    
    my ($gene, $cols) = split /\s*\|\s*/, $_, 2;
    
    # Here columns are not stored in an array, but in a string
    # if the data is:
    # gene 3 | value 3.1 | value 3.2
    # then $cols would be:
    # 'value 3.1 | value 3.2'
    $data{$gene}{$table} = $cols;
    $table = '' if eof;
}
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Does this explain better? I suspect that some of your question is about $ARGV and where that comes from and the input (angle) operator (while (<>)) and reading files from @ARGV.

Chris