If I change the assignment to @my_name to

@my_name=("Acidovorax", "sp.", "JS42");
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the program fails to find a single match. I doubt this is want the OP wants. The problem is that you're using different "word" definitions for "my_name" and "db_name". Either split both on whitespace, or both on \b, or it becomes trivial to find examples where the search is going to return the wrong answer. For instance:

use 5.10.0;
use strict;
no warnings;

my @names    = ("Acidovorax JS42", "Acidovorax sp. JS42");
my @db_names = ("Acidovorax sp. JS42",    #data base of names in array
                "JS42 Acidovorax sp.",
                "JS42 sp. Acidovorax",
                "Acidovorax JS42",
                "JS53 sp. Acidovorax",
                "JS42 sp. Axidovorax",
                "JS42Acidovorax sp. "
                );

foreach (@names) {
    say;
    my %name; @name{+split} = ();
    foreach (@db_names) {
        my %copy = %name;
        delete $copy{$_} for split;
        say "\t$_ is ", (keys %copy ? "not a " : "a "), "match";
    }
}
__END__
Acidovorax JS42
    Acidovorax sp. JS42 is a match
    JS42 Acidovorax sp. is a match
    JS42 sp. Acidovorax is a match
    Acidovorax JS42 is a match
    JS53 sp. Acidovorax is not a match
    JS42 sp. Axidovorax is not a match
    JS42Acidovorax sp.  is not a match
Acidovorax sp. JS42
    Acidovorax sp. JS42 is a match
    JS42 Acidovorax sp. is a match
    JS42 sp. Acidovorax is a match
    Acidovorax JS42 is not a match
    JS53 sp. Acidovorax is not a match
    JS42 sp. Axidovorax is not a match
    JS42Acidovorax sp.  is not a match
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