I am planning on making a custom engine for my game, and I have 2 questions.

1. Does it have to  be a recognizable engine like Unity or Godot?

2. Does it have to be made in the game development time frame as well?

In case you're curious, the engine is simply focusing on drawing them and storing data right now. :D

Yes, you can use a custom engine, since there are no limitations on engines or tools. And an engine certainly counts as general purpose code, so you’re allowed to build it before the jam. Good luck!