[Anthony]

To have any type able to give birth to any other type, either everyone needs genetics for all of the types and something activates or deactivates specific combinations (you only need three on/off controls to generate 8 types; to get it down to seven types you might make one combination lethal).

For example, you might have:
Zero activated genes: baseline mundane.
One activated gene: one of three slightly exotic races.
Two activated genes: one of three highly exotic races.
Three activated genes: dead (or other problem).

Now, for inheritance, I would probably make the increased odds of the same race be strictly maternal, because if the trigger is environmental the father is remote enough that you probably don't get a meaningful contribution, and it also makes my math a bit easier.

Okay, if inheritance is entirely random, we'd expect a 12.5% chance of matching the parent (but one type is excluded, so overall 14%). If activation is instead a 60% match on the mother, we get a basic 21.6% chance of being the same, but a 6.4-14.4% chance of the excluded combination, for a net among survivors of a hair under 24%.