The answer: Swanstone
How I found it: I got lucky. I just grabbed it and started taking it to local big-box places (I assumed it was a big-box as the materials in this house aren't necessarily 'high end').
I stopped at Lowes and the initial guy in the kitchen section had no clue, but I was fortunate that a curious co-worker popped his head over the counter, thought for a minute, then declared it 'Bermuda Sand' from 'Swanstone'. We found the swatches and, sure enough, he was right-on. So kudos tot he very helpful Lowes employee.
Apparently, these type of shower systems are just large panels of solid surface material that is glued up on the wall. They idea is that the panels abut, then caulked. The corner cove is more of an 'extra' piece that can be used if desired. The underlayment SHOULD be your standard tile surface...water proof cement board that is then sealed with some sort of water sealant. However, I don't know that in this case, so this particular piece seems to be a rather critical for my shower. So, glad we were able to find it.
The challenge for future people reading this is that I guess the solid surface companies are many, and the patterns all tend to be unique to each company, so it is a bit of a needle-in-a-haystack hunt trying to find these.
For contractors, I'd like to suggest leaving a note on the inside of a bath or kitchen vanity with the manufacturer's info. That'd be handy. ;)
You are absolutely right that the floor will move independently of the walls. The movement when an adult steps in is not as much as a full bath of water, say, but it is still enough that you need a flexible seal of some kind to avoid water pooling and mould growing.
I asked a similar question which @ChrisF answered very well which suggested that caulk should be flexible enough to cope. That is what I did, and so far I have had no leaks (I have taken the panels off to check every now and then)
We installed our shower base – round corner, and tiled our walls. Now our glass shower doors on steel frames don't fit on the base. It's too small and doesn't have enough room to sit inside of the base. It's off by about 1 inch combined from both sides. I don't think we took into account the walls. Is there anything we can do to fix this? 
Best Answer
Unfortunately while your shower door and base look nice, the engineering on them is whack. Just looking at where the door should sit... its not even flat. I am guessing (hoping) that there is a bevel to the bottom of the door that makes it "fit" but just a flat surface works better.
I have put in 10s of these things. On all of mine I have had the OPPOSITE issue. I have to build out to a depth so the door isn't inside.
What could you have done better? Well you should have set your base up and put the door on to test out the dimensional fit before installing the backer and tile. If you are off by that much there is no way this thing will fit.
(pre) Solution? You could inset the wall compared to the rest of the room by using thinner concrete backer or possibly make the rest of the room thicker... You could also push the base out by maybe adding a backing to it. It is too late to try these things as you would need to rip out your base and redo plumbing a bit.
(post) Solution? You cannot cut out a space for your door. I have no idea why that is being suggested in the comments. I have never seen anyone with any tool that could cut out tile in a straight line. Sure you can cut it out. Just like I can spray paint my car myself when I need a paint job. Expect to use lots of caulk after and for it to look really bad and possibly leak.
Me guesses... I would guess that the manufacturers run sheet on this install is doing it with a PVC wall system. I am guessing that they suggest that the walls go right on the 2x4 framing. Then your door fits. There really isn't another way given your details are correct. (do you have the manufacturers install instructions for this? it is common for these bases/doors to be sold separate from a wall system as they can give you a variety of choices on the wall system.)
How do we salvage this? The only solution without ruining the integrity of the tile work you have done and possibly ruining the waterproof system is to cut the door to make it fit. Cutting the metal is not that big of a deal. The cuts should be hidden for the most part and the side pop back in after. If you can only cut one side obviously that is half the work but I am not there and doubt it. The problem is you probably cannot cut the glass as it should be tempered. Meaning local glass shop will have to send out two new tempered sheets. So your cost of repairing this right is really just those two new sheets (100-150 in my area).