im sorry i keep just posting more comments! I just finished the first moon and it seems i was kinda wrong. The upgrades do save but the buttons to buy them dont. So you have to rebuy all the ones youve gotten already. Also I am currently stuck here and it doesnt send me back to the map. but trying to leave says it will loose my save. Send help lol 
You are correct, the updates are saving, but the actual buttons re-appear on game over and maybe on a new level. For now, you can just ignore them, as long as you don't mind seeing them in the shop. I can definitely fix this. I'm still trying to replicate the other issue where the level doesn't end.