DL, I don't know if you've ever seen a screensaver called VR Man's Aquarium, but it had exactly the kind of chase cam feature you describe. The screensaver featured a completely 3D freshwater aquarium that was in a small room.
You could move the camera around manually, right into the aquarium itself if you want (and the lighting changed accordingly, to slightly more subdued, when you were in the aquarium), or you could move outside the aquarium into the room, and eventually zoom out far enough into the 3D room so that the aquarium was a relatively small object against the far wall.
You could also choose to follow any particular fish in the tank. I found this more nausea-inducing than enjoyable. Those fish moved fast!
You could also feed the fish, though the fish didn't actually "eat" the food. They just sort of herded up to it, and the food "disappeared." Their mouths never opened.
You could also make your own textures for the gravel and background of the tank. I impressed my family by firing up the tank with a picture of my niece and nephew as the tank background.
The main problems with this screensaver were:
1. The fish were nowhere near as realistic as those in the Sachs aquarium. Not even remotely close.
2. Can you say clipping problems? This screensaver had tons of them.
3. VR Man's screensavers all had the same annoying flaw: The camera angle cuts were random and were sudden, not smooth. So the camera would be floating over the tank for a while, then -- cut -- you're following a rapidly swimming fish, then -- cut -- you're on the gravel looking up at the fish, then -- somebody give me some Tylenol.
Nonetheless it was the first aquarium screensaver that actually impressed me. Nowhere near the Sachs one, but pretty good for its time.