Is it possible to use Bluetooth serial and BLE client simultaneously in ESP32?
Yes, according to this GitHub issue it is possible to use classic Bluetooth and BLE together.

GitHub user @beegee-tokyo who raised that issue suggests to initialize the Bluetooth and BLE services in the following manner :-

Init BLE device =&gt; BLEDevice::init(apName);
 Init Bluetooth Serial =&gt; SerialBT.begin(apName);
 Then do the rest of the BLE initialisation like
  - create server
  - add callbacks to server
  - add characteristics
  - start server
  - start advertising

He has also posted his finding on his blog and wiki and has also shared the following code snippet :-

// Initialize BLE
BLEDevice::init(apName);
BLEDevice::setPower(ESP_PWR_LVL_P7);

if (!SerialBT.begin(apName)) {
    sendDebug(debugLabel, errorLabel + digitalTimeDisplaySec() + " Failed to start BTSerial", false);
}

// Create BLE Server
pServer = BLEDevice::createServer();

// Set server callbacks
pServer-&gt;setCallbacks(new MyServerCallbacks());

// Create BLE Service
pService = pServer-&gt;createService(BLEUUID((uint16_t)SERVICE_UUID),20);

// Create BLE Characteristic for Alert
pCharacteristicNotify = pService-&gt;createCharacteristic(
    BLEUUID((uint16_t)NOTIFICATION_UUID),
    BLECharacteristic::PROPERTY_READ     |
    // BLECharacteristic::PROPERTY_INDICATE |
    BLECharacteristic::PROPERTY_NOTIFY
);

// add other characteristics

// Start the service
pService-&gt;start();

// Start advertising
pAdvertising = pServer-&gt;getAdvertising();

pAdvertising-&gt;start();