Yes, the generations do have an effect but it is different for the wave 1 (Comes in blue, green, and white) and wave 2 (Comes in black, pink, and purple) Tama-Gos. For example, you get Lovelitchi on an odd generation on a wave 1 Tama-Go, but on an even generation on a wave 2 Tama-Go.
I know all the generations they have to be, so I might make a growth chart sometime. You can PM me if you want to know what generation your Tama has to be for a certain character if you like.
There are two characters, one for each generation number, for each level of care. For example, on Wave 1, Mametchi and Kuromametchi are both gained by perfect care but Mametchi is only obtainable on odd generations and Kuromametchi is only obtained on even generations (on the Wave 2 Tama-Go series, it's the other way around--Mametchi is on even generations and Kuromametchi is on odd generations).