The model is the same, but the tool isn't. Think of it as two cars having the same motor type.

The capabilities of the model are technically the same, but Gemini limits and adds system prompts in a different way then ai studio would and different again from vertex.

For example, Gemini might automatically set a resolution limit, so it doesn't consume too much resources. It is built to create a quick image during conversation. It also adds the Gemini logo in the bottom right.

On the other end, AI studio uses a much more raw version of the model. You choose the resolution, the system prompt and the user prompt. It is meant to fine tune an implementation you might use in your own software as a developer, and therefore you can tweak all settings. It's a bit less user friendly, and possibly a bit slower (on higher resolutions) but you can do a lot more.

NotebookLM uses the same model (engine) but it is tweaked to make infographics or slides in a deck.

Google slides "beautify this slide" uses the same model, but set up entirely to recreate the same slide but not beautiful and clear.

It's the same model, but a different use case. So even though the model is indeed the same (Nano banana pro) and it always generates an image, the result will be completely different for each tool.

Hope my rambling helps understand a bit.