It goes in fours. (By the way, the links that I provide are just links to my log when the first occurrence of each "unlock event" takes place. I go into some more detail in those posts so feel free to check it out if you'd like!
)
The
default icon background:
No change in icon background is achieved after 4 unique characters are obtained but a new location, the
Henshinjo, is unlocked.
A new location,
Patchi Onsen, is unlocked after getting the TamaTomo stamps for 8 different characters, and the Fruits icon background is unlocked:
Melody Land is unlocked after earning TamaTomo stamps for 12 characters. The Melody background comes along with it after the initial visit to Melody Land:
The
Concert Hall opens up after 16 unique characters. No new icon background.
After getting TamaTomo stamps for a total of 20 different characters, the
Waru Cafe opens up. The Waru icon background is available after this:
A new location,
Uranai No Yakata, unlocks after 24 unique characters are obtained. There isn't a new icon background that comes with this new location.
The
Gotchi School is unlocked after getting 28 different characters. Again, no new icon background.
Finally, obtaining TamaTomo stamps for all 32 characters will unlock the
Gotchi King's Castle, as well as the accompanying King icon background: