Anyway - LIKE is fail as well

Sorry, but that is not an adequate problem description. Please see How do I post a question effectively? and I know what I mean. Why don't you?, and provide a Short, Self-Contained, Correct Example, like the following. Note how it is exactly the code you posted, except that it works fine for me.

use warnings;
use strict;
use feature 'say';
use DBI;

my $dbh = DBI->connect(
    "DBI:mysql:database=testing;host=127.0.0.1", $ENV{USER},
    'barfoo', { RaiseError => 1, AutoCommit => 1 });
$dbh->do('DROP TABLE IF EXISTS texts');
$dbh->do(<<'ENDSQL');
CREATE TABLE texts (
    string VARCHAR(256) 
);
ENDSQL
$dbh->do('INSERT INTO texts (string) VALUES ("/a/b/c");');

# ---
 my $string = '/a/b/c';
 my $req = qq{ SELECT * FROM `texts` WHERE `string`=?};
 my $sth = $dbh->prepare($req);
 my $row = $sth->execute($string);
 say "row = $row";
# ---

my $sth2 = $dbh->prepare(
    q{ SELECT * FROM texts WHERE string LIKE '%/a/b/c%' });
$sth2->execute;
say "row = $row";
[download]

This prints row = 1 twice, as expected.

This is how I spun up a fresh test MySQL instance. To change the server's port number, change the -p3306:3306 to something like -p13306:3306 in the docker command, add the argument --port=13306 to the mysql command, and add the argument ;port=13306 to the DBI connection string in the Perl code.

$ docker run --rm --name testmysql -p3306:3306 -e MYSQL_ROOT_PASSWORD=foobar -d mysql:5
# wait a few seconds for it to start
$ echo "CREATE USER '$USER' IDENTIFIED BY 'barfoo'; CREATE DATABASE testing; GRANT ALL PRIVILEGES ON testing.* TO '$USER';" | mysql --protocol=TCP --user=root --password=foobar
# after testing:
$ docker stop testmysql

Several smaller edits to add more information.