I have modified and uploaded some sketches from "ArduinoBLE" examples to my nano 33 ble board. I can control the app from "low level apps" like nRF Connect and LightBlue. But I want to control i my nano board from an android app with a Game pad or Remote control UI.
You are a being rather vague with describing your problem. The ArduinoBLE library starts with two options "Central" or "Peripheral". You haven't said which example group you chose, but it's easy to surmise that you tried some Peripheral examples. My ArduinoBLE library provides 3 options for LED and 1 for battery monitor. So which did you try? Once again as you mention using a "low level app" can presume an LED example.
Putting that aside, I am trying to visual your architecture. Are you wanting your Nano to connected to an Ardroid app, which in turn is connected to a Game Pad or Remote control UI?
From the looks of things, if your Game Pad or Remote control UI is using Bluetooth EDR (2.0) then you would be right to use an Android phone/tablet as the bridge.
The problem I am trying to solve is to use arduino nano 33 ble as a bluetooth remote control for a toy-car.
This statement says the opposite. In theory, if your remote control toy-car was using a bluetooth low energy device then you could connect the nano BLE device to the car directly by defining the nano as a "central" device.
If we assume that the car uses "old style" Bluetooth, then I would suggest you look at MIT App Inventor as you can readily build your app fairly quickly as you can use both Bluetooth standard and Bluetooth Low Energy.
If I implement some predefined characteristic like "Human interface device 0x1812", do I also need to implement specific services?
If using MIT App Inventor as the means to make the bridge then you have to freedom to use what ever you want. The simplest and quickest is to use the existing custom service provided by the nano. There is no need to try and use the HID 0x1812 option here.
However, if the remote control device happened to be using Bluetooth Low Energy then you need to read the specification to find out what services it is using. You can of course try using your "low level apps" for this as it would reveal those details. Word of caution though; if the remote control device was using encryption (pairing) it make this option difficult if a password or pin code was required.
On the other, the simplest method might be just to purchase another nano (one for car and one for remote control).