I also don't have co-op mode and I cannot build cabins :(
Between not having multiplayer that app icon, that's definitely a copy of the mobile version. That's interesting, thinking back there actually may have been a brief window when the mobile version was available to download on Macs, however, that version doesn't offer multiplayer even now and unfortunately does not appear to be available anymore on Mac. Apologies for doubting this before, but I believe this only the second time I've seen someone have the mobile version on a Mac, and it wasn't until I saw that screenshot you sent that I remember having seen this before.
I think at this point you will have to purchase the game through Steam or GOG, but you would have to purchase one of those version anyways to get multiplayer. The good news is that your saves from that version you have no should still remain available as long as you have it installed, and should be transferable to the Steam/GOG version. If you open Finder, there should be a folder called "On My Mac" in the sidebar on the left and within that there should be a Stardew Valley folder with your saves. Let me know if not, this is more of a guess based on where saves are located on an iPhone/iPad.