Repeated DB connection at max user connections
You code has several problems. Specificaly it breaks at least two of the SOLID principles, namely single responsibility and dependency inversion principles.
Single responsibility is violated because the class is responsible for:

connecting to db
retrying connection
whatever other methods provided by the class you did not show us

Dependency inversion principle is violated because the class uses hardcoded values (although stored in instance properties).
Let me first separate out the first responsibility. We don't really need OOP for that, so let me do it in FP style.
function createMyPdo(): \PDO
{
  $dsn = ...;
  $user = ...;
  $password = ...;
  $options = ...;
  return new \PDO($dsn, $user, $password, $options);
}

This function now opens a connection with hardcoded credentials. Although the credentials are hardocded and we would better pull those values off of maybe environment variables. It at least has just one responsibility and that is to open the specific database connection.
Now lets implement the retry logic as a separate thing, that only depends on something that creates PDO instance.
function createPdoWithRetry(callable $factory, int $attempts): \PDO
{
  $attemptsMade = 0;
  while ($attempts &gt; $attemptsMade) {
    try {
      return $factory();
    } catch (\PDOException $e) {
      if ($e-&gt;getCode() === 1040) {
        \usleep(\pow(2, $attempsMade) * 10000);
        ++$attemptsMade;
      } else {
        throw $e;
      }
    }
  }
  throw new \RuntimeException(&quot;All $attemptsMade retry attempts exhausted.&quot;);
}

Now I can easily open the connection with retry logic
$pdo = createPdoWithRetry(fn () =&gt; createMyPdo(), 20);

And I can also open a different connection using the same thing:
$pdo2 = createPdoWithRetry(fn () =&gt; createOtherPdo(), 20);

This approach still has some caveats:

where createMyPdo could only throw PDOException, createPdoWithRetry can also throw a generic RuntimeException.
the type of the $factory parameter has a very vague type, that we have to describe in a docblock and cannot be enforced by PHP runtime.
createPdoWithRetry knows error code 1040 which is mysql specific, but the knowledge that mysql is used is only known to the createMyPdo function.

Let's make it better by going back to OOP style and introducing an interface for the connection factory, including specific exceptions to be thrown (which will abstract away the 1040 speific mysql error code).
class DatabaseConnectionException extends \Exception
{
}

class TooManyConnectionsException extends DatabaseConnectionException
{
}

interface DatabaseConnector
{
  /**
   * @return \PDO
   * @throws DatabaseConnectionException
   * @throws TooManyConnectionsException
   */
  public function createPdo(): \PDO;
}

As you can see I added some @throws annotations to:

inform implementors of the interface which exception they can throw
informs consumers of the interface which exceptions they can catch

Although PHP cannot enforce the actual type of exceptions thrown from the implementations, documenting the exceptations makes it less likely that someone will implement it wrong.
Now lets make an implementaton for our specific connection:
class MyDatabaseConnector implements DatabaseConnector
{
  public function createPdo(): \PDO
  {
    $dsn = ...;
    $user = ...;
    $password = ...;
    $options = ...;
    try {
      return new \PDO($dsn, $user, $password, $options);
    } catch (\PDOException $e) {
      if ($e-&gt;getCode() === 1040) {
        throw new TooManyConnectionsException($e-&gt;getMessage(), (int) $e-&gt;getCode(), $e);
      } else {
        throw new DatabaseConnectionException($e-&gt;getMessage(), (int) $e-&gt;getCode(), $e);
      }
    }
  }
}

Now maybe you see why the credentials should not be hardocded, because the same code would work for any credentials. Hardcoding the credentials, we are doomed to repeat some code if we want to open a different connection.
But I leave that up to you to find your way to the dependency inversion principle. Basically constructor of a class should accept things that the class instances need to know, rather then having the instances deciding on their own what those values should be (ie. pass the credentials through constructor, rather then hardocing the credentials withing the class).
Anyway, to implement the retry logic we will use the same interface:
class RetryingDatabaseConnector implements DatabaseConnector
{
  private DatabaseConnector $factory;
  private RetryDelayStrategy $delayStrategy;
  private int $maxAttempts;

  public function __construct(DatabaseConnector $factory, RetryDelayStrategy $delayStrategy, int $maxAttempts)
  {
    $this-&gt;factory = $factory;
    $this-&gt;delayStrategy = $delayStrategy;
    $this-&gt;maxAttempts = $maxAttempts;
  }

  public function createPdo(): \PDO
  {
    $attemptsMade = 0;
    while ($this-&gt;maxAttempts &gt; $attemptsMade) {
      try {
        return $this-&gt;factory-&gt;createPdo();
      } catch (TooManyConnectionsException $e) {
          ++$attemptsMade;
          $this-&gt;delayStrategy-&gt;wait($attemptsMade)
      }
    }
    throw new TooManyConnectionsException(&quot;All $attemptsMade retry attempts exhausted.&quot;);
  }
}

As you can see I have also extracted the delay logic to a separate interface, because exponentialy growing gaps between attempts is just one of many possible strategies. And again, you don't want to implement a new connector that would repeat a lot of the logic just to change to delay strategy.
interface RetryDelayStrategy
{
  public function wait(int $attemptsMade): void;
}

class ExponentialRetryDelayStrategy implements RetryDelayStrategy
{
  private int $coefficient;
  private int $base;

  public function __construct(int $coefficient, int $base = 2)
  {
    $this-&gt;coefficient = $coefficient;
    $this-&gt;base = $base;
  }

  public function wait(int $attemptsMade): void
  {
    \usleep(\pow($this-&gt;base, $attempsMade) * $this-&gt;coefficient);
  }
}

And finaly some consumer code. A controller that opens a database connection (to do something with it) and gives a user friendly messages if the connection fails and a more specific one if the connection fails specificaly because of too many open connections.
class MyController extends BaseController
{
  private DatabaseConnector $connector;

  public function __construct(DatabaseConnector $connector)
  {
    $this-&gt;connector = $connector;
  }

  public function myControllerAction(Request $request): Response
  {
    try {
      $pdo = $this-&gt;connector-&gt;createPdo();
    } catch (TooManyConnectionsException $e) {
      return $this-&gt;send(503, &quot;Too many database connections.&quot;);
    } catch (DatabaseConnectionException $e) {
      return $this-&gt;send(503, &quot;Database unavailable.&quot;);
    }

    // do something with PDO here

    return $this-&gt;send(200, &quot;All done&quot;);
  }
}

Now, you see the controller just depends on a DatabaseConnector. It doesn't really care if and how many times the connector will retry. It just cares that it can fail in a specific (TooManyConnectionsException) or an further unspecified (DatabaseConnectionException) way or, if it succeeds, it returns a PDO instance.
And it will work no matter what connector you pass to it, as long as the interface is implemented correctly. And so you can change the implementation without touching the controller code. Again, thanks to single responsibility a dependency inversion principles.
No matter which is used in the DI setup
$myConnector = new MyDatabaseConnector();
// or maybe if we pull it to envs
// $myConnector = new MyDatabaseConnector($_ENV['DB_DSN'], $_ENV['DB_USER'], $_ENV['DB_PASS']);
$connector = $myConnector;

or
$delayStrategy = new ExponentialRetryDelayStrategy(10000, 2);
$retryingConnector = new RetryingDatabaseConnector($myConnector, $delayStrategy, 20);
$connector = $retryingConnector;

the controller code stays untouched
$controller = new MyController($connector);
...
$controller-&gt;run();

Also notice, how I avoided setting an errorCode and checking it in the controller. Instead we just throw exceptions. This forces the caller to actually handle the error or let it bubble up the stack and eventually stop execution of the program.
This is the prefered behaviour because if you would forget to check the error code, it really does not make sense to try to send sql queries over a &quot;not opened connection&quot;.
Btw, if you are interested in design patterns, you can notice that:

DatabaseConnector and its implementations are a factory method pattern
RetryingDatabaseConnector and the interface follow the decorator pattern
RetryDelayStrategy and its implementations are the strategy of the strategy design pattern and RetryingDatabaseConnector is the strategy consumer