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.

if ($_ = (m/^[\$]\w/)) {
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Solution pointed out below. This is an assignment (using =) clobbering $_ with an empty array the true/false result. An assignment of an undefined thing evaluates to false every time. Similar to if ($_ = ()) or if ($_ = undef).

Update - because I was curious:

perl -MO=Deparse,-P -e 'if($_ = (m/foo/)) { print 'hi' }'
if ($_ = /foo/) {
    print hi $_;
}
-e syntax OK
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A little more insight.

update 2- corrected ty

This is an assignment (using =) clobbering $_ with an empty array. An assignment of an undefined ... Similar to if ($_ = ()) or if ($_ = undef).

$_ is clobbered with the true/false result of evaluating the match of the regex against $_:

c:\@Work\Perl\monks>perl -wMstrict -MData::Dump -e
"print 'Enter a string and I will determine the data type: ';
 chomp($_ = <STDIN>);
 dd 'A:', $_;
 ;;
 if ($_ = (m/^[\$]\w/)) {
    print qq{\n'$_' is a scalar data type\n};
    }
 dd 'B:', $_;
 ;;
 if ($_ = (m/^[\@]\w/)) {
    print qq{\n'$_' is an array data type\n};
    }
 dd 'C:', $_;
"
Enter a string and I will determine the data type: $scalar
("A:", "\$scalar")

'1' is a scalar data type
("B:", 1)
("C:", "")

c:\@Work\Perl\monks>perl -wMstrict -MData::Dump -e
"print 'Enter a string and I will determine the data type: ';
 chomp($_ = <STDIN>);
 dd 'A:', $_;
 ;;
 if ($_ = (m/^[\$]\w/)) {
    print qq{\n'$_' is a scalar data type\n};
    }
 dd 'B:', $_;
 ;;
 if ($_ = (m/^[\@]\w/)) {
    print qq{\n'$_' is an array data type\n};
    }
 dd 'C:', $_;
"
Enter a string and I will determine the data type: @ra
("A:", "\@ra")
("B:", "")
("C:", "")
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---

Give a man a fish:  <%-{-{-{-<

ty fixed