I have a 97 A4 2.8. When I turn it on, it says BOSE.

CAN SOMEONE PLEASE TELL ME WHAT THE HELL MY STEREO IS CALLED?

And maybe what all the OTHER stereos are called and which cars they came in.

the actual HU name is on the flap for the cassette/tape.
You may have the Bose upgrade which means the HU is the same, but has a flatter EQ (turned on/off by earthing the Bose pin on the back of the HU), and then you'll have a seperate Bose amp and Bose speakers.
If you get a dull woolly sound from the system then it's Bose. If the treble & bass actually work then it's non Bose. Non Bose system has just a rear amp, which is bolted to the rear left deck speaker.
This we site lists all the models<ul><li><a href="http://www.ibiblio.org/tkan/audi/radios/radios.html">http://www.ibiblio.org/tkan/audi/radios/radios.html</a</li></ul>

I like it and gladly pay extra for it.

You know how much you have to pay to get custom equalization for a car (meaning mics are placed in the car and each speaker is then equalized to give a flat (desired) response so that there is no attenuation of the sound.)

Oh, I'm a huge Bose fan and think that they make great stuff but I can admit when there is a better sounding system or if a particular Bose unit isn't that great.

I *think* I have the non-Bose because I see an amp bolted on the left in the trunk under where the left speaker is. Is this correct?

In this case, the BOSE lights up on the screen but there is really nothing Bose about it?

This isn't an opinion it's a fact.
Their Wave system sounds good for a clock radio but hopeless as a ghetto blaster.
Similarly, their Acoustimass system is OK as a home HiFi but far superior sounding systems are available for a fraction of the cost.
Their 801 PA speakers only sould good when taking into account their size. If they were twice the size you'd think they sound pretty average.
Anyone who thinks Bose sounds superb just hasn't heard a decent sounding system, it really is as simple as that. There's even web sites dedicated to their mediocrity.
"No highs, no lows it must be Bose"
Listen to the Volvo DynAudio upgrade if you want to hear a decent stock system in a car.

But yes the Bose on the display just means the HU has been swapped out at some point, and the replacement was from a Bose equipped Audi.

I can't imagine why the stereo would have been replaced. The car has low miles and was owned by an older, wealthy gentleman.

This should help.

http://www.ibiblio.org/tkan/audi/radios/radios.html