我对PHP和MySQL感到头疼

I have this problem:

function search_by_name($mysql, $name, $lastname)
{
    $query = 'SELECT idKlienci FROM Klienci WHERE Imie = "' . $name . '" 
              AND Nazwisko = "' . $lastnem . '"';
    $result = $mysql->query($query);
    $row = mysqli_fetch_array($result); // I want to get the ID' of table `Klienci1`
    // and here i don't know how many dimentions have this array
    echo $row[0][0]; // prints nothing
}

MYSQL doesn't use double quotes for strings, it uses simple quotes:

$query = "SELECT idKlienci FROM Klienci WHERE Imie = '$name' AND Nazwisko = '$lastname'";

You can also add the variable directly into PHP strings when you're using double quotes, without the need to concatenate them.

$lastnem != $lastname

So change the query to use the correct variable name

$query = 'SELECT idKlienci 
            FROM Klienci 
            WHERE Imie = "' . $name . '" 
              AND Nazwisko = "' . $lastname . '"';

To make this kind of code easier to read you can also make use of the fact that variables in a double quoted string are automatically expanded. Which make this easier to read and therefore debug.

$query = "SELECT idKlienci 
            FROM Klienci 
            WHERE Imie = '$name' 
              AND Nazwisko = '$lastname'";

$result = $mysql->query($query);

// use mysqli_fetch_assoc() then you get only one assoc array
// so you can use named parameters to the array.
// the names will match the column names in the table

//$row = mysqli_fetch_array($result);
// also mysqli_fetch_assoc() only returns one row at a time
$row = mysqli_fetch_assoc($result);

// a row is always one dimensional so do 
echo $row['id']; 

So if you have more than one row in the resultset of your query you have to get the results in a loop

$result = $mysql->query($query);

while ( $row = mysqli_fetch_assoc($result)) { 
    echo $row['id'] . '<br>'; 
}

Now you should see both rows

Your Argument name $lastname & used variable $lastnem name are not same.

try this :

 $row = mysqli_fetch_array($result); 
    echo $row[0];