Long answer: The $_ is the default topic. You don't need to mention it in many contexts, matching a regex being one of them. So, you can shorten the code to

if (/^[\$]\w/) {
    print "\nThat is a scalar data type\n";
} elsif (m/^[\@]\w/) {
    print "\nThat is an array data type\n";
} else {
    print "\nData type not found\n";
}
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When you use =, it changes the contents of $_, so after the first condition, it contains the result of the regex match, not the original user input.

Also, remove the my from $_. Lexical $_ was deprecated and is not supported in modern Perl versions. Use local if you want to incorporate the code into a larger codebase.

Update:

Moreover, you can also use a hash from sigils to data types, no regex and if/else needed:

print "Enter a string and I will determine the data type: ";
chomp(local $_ = <STDIN>);

my %sigil2type = ('$' => 'scalar',
                  '@' => 'array',
                  '%' => 'hash',
                  '*' => 'GLOB');

my $sigil = substr $_, 0, 1;
my $type = $sigil2type{$sigil} || 'unknown';
say "\nThat is a $type data type.";
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Update2: The $_ will be populated with the match result, not with the regex. Thanks AnomalousMonk.