I can't see anything holding this cartridge in, but it doesn't want to pull out. I am 99% sure that it is a Moen, since it matches the deck mount tub filler, which I know is a Moen.
Have you tried putting the screw in the handle and screwing the screw into the cartridge , so you have something good to pull on. sometimes there just hard to get out.
No, I didn't think of that, but I did use needlenose to try to pull it out. When I am there tomorrow rebuilding the deck mount valve I will give that a try.
I don't think it's Moen. It looks like Delta. It's the same a the Delta Delex. There is a nut down there that needs to be removed to pull the stem. You need a very thin wall socket. Delta shipped a tool for that with the faucet.