How many types of dog houses are there in plastic design engineering?

There is no such thing, different types of doghouses(DH).
DH is a b side feature used for assembly fixation (or to increase the height of any features) generally with biw cut-out, with a plastic clip on it.
some times featues like locator, snaps, ect will be fecilitated on DH, because if there height is too much, the feature will fail.
so by doing that, features height is decreased and strength increased
And, DH is used for different purpose in diff locations depending on situations.
here is the DH in picture used for both location and assembly.
Hopes helpful:)

It depends on your customer's standard. in my experience they have types of "keyhole", shape of doghouse that also depends on the location on the product and how/what direction it will be assemble. (ex. car interior parts)