Question:
I just paid a nice reduced price for what I thought would Skyrim Anniv Edition. The only download name I can find on my PC that links to my order date says that I have Skyrim Special Edition. The GOG page shows that I paid for Anniversary Edition.
Are these two the same??
I used GOG to download. Did I miss something?

You will know what you got after installing.

People have a habit of confusing this and to be fair Steam, GOG etc don't really help. For instance, in the GOG app UI its labelled as SE. I'd recommend checking your file pathways wherever you have stored them instead. If that says AE at the end of Skyrim, you got what you paid for.