I would classify the game as a arcade/puzzle. 8bit graphics, 2D. Probably a platformer, although I'm not sure if in some levels there was a top-down view. Singleplayer.
I will describe here one of the levels that I remember the most, which means it had to be the first or one of the first:
The level fit on one screen. A separate corridor occupied the bottom of the screen. In it, on the left, below the player, the cat/leopard/panther/lion /... started the level. At the other end of the corridor on the right was a cage, and between there were several gates/doors/locks. The cat ran left and right between the blocks. The goal of the level was to lock the cat in the cage by opening the locks in front of him and closing those behind him. This was done by operating blockades from above. Opening only required an approach. Closing probably some kind of jumping on it. There were some creatures on the board chasing the player. They started on the right. You couldn't touch them. I think they also disturbed (randomly?) opening/closing the locks. On the left and right there were some typical platform constructions. Entrances in the form of ladders or ropes or something like that. I don't remember if they had any application other than allowing the outmaneuvering of the opponents (maybe falling on blockades to close them? - this could be a false clue). The player-controlled character was humanoid.
Quick sketch:
Later levels repeated the motif of enclosing the animal in a cage (maybe even this was always the goal). The layout was of course different. It was possible to touch the cat, which ended in failure. I also remember something like collecting items (keys/fruit/bananas? Used as bait?). There were also levels larger than one screen. Other animals appeared, too. I remember a lot of greenery on one of the lvls.
The difficulty was usually to find a way to enclose the animal in the cage and implement the plan - hence both puzzle and arcade. When I played this game, I certainly didn't know the language (English / Japanese?), so it's quite possible that I wasn't able to catch all the mechanics and my description is a bit different from the heart of the game. However, I was able to pass several levels, so the game had to be simple, accessible to children.
I went through a few "All NES Games" videos, but I didn't find it, suggesting that this game could be a hack or part of some kind of 1000 in 1 medley. But I could simply just overlook or not even recognize it in the video.
Anyway, I am asking for any help and possible associations.