This app was mentioned in 2 comments, with an average of 8.50 upvotes
Boku to Wanko:Doggie Collector. It's from a different developer than the one that made Neko Atsume, but it looks like they went to great lengths to make the game look and feel similar to Neko Atsume.
Mushroom Garden is pretty similar.
There's a dog copycat of Neko Atsume
I've had my eye on AbyssRium but haven't had a chance to play it.
I was also looking at Slime Evolution because it looked similar to Neko Atsume but with an added battle element.
KleptoCats also looked good.
Viridi is a zen succulent planting sim. It gets kind of boring though because there's not much to do.