I've never learned a term for this and I can't think of a direct analogy in English.
What about -gate, to mean memorable scandal? It's not really the same, I guess. This kind of word formation is much more productive in ASL (and in sign languages generally) than in English (and spoken languages generally). I've listed the examples you gave, and a few more:
[T]rigonometry, [A]lgebra, [C]alculus, ...
[Y]ellow, [B]lue, ...
[Family, [G]roup, [C]lass, ...
[L]aw, [R]egulation, [E]thics, ...
Is there a recognized way of referring to these groups of words?
These are frames. A frame is, in all the examples up there, the way the hand moves, plus non-manual features, but is not a complete sign (and so does not have a "complete" meaning). The sign is completed by adding a handshape. The examples in the Wikipedia link are completed by adding the handshape from another ASL word, but the examples you and I gave are completed by adding the handshape from [the initial letter of] an English word.
I've been taught that unnecessary initialization is sometimes seen as undesirable in the Deaf community. Does that still apply to words like this, where initialization is the only way of distinguishing them?
It doesn't apply here, because without the initial, you'd just be left with the "frame", which is impossible to produce in isolation!, and which would not mean anything by itself anyway. The sign you know for family, really is the entire sign.