Ok, so the -id in these names (Didonids; cf "Barcids;" your Nasrids; cf Athenian Alkmaionids, Eteobutidae, etc) is not from Semitic at all; it is from Greek. There is a Greek suffix -is, genitive -idos and therefore with a stem -id that means loosely "pertaining to, born from." We have no idea where this particular formation came from, and the delta in particular is problematic. Some have attributed it to noun examples (e.g. in Latin, classis or lapis) and others to formations in Sanskrit like -īs, gen. -iyas.

This is a pretty obscure class in Greek, but it is a "productive" suffix, meaning that subsequent languages used it in that context, especially when referring to families or dynasties or the like. In English, we use it rarely elsewhere, except oddly for naming meteor showers: the Perseids, the Leonids. It is also common in scientific notation, e.g. "hominid", which is a Latin root (homo plus this Greek suffix). The more common generic Greek adjectival suffix is -ikos, which generates hundreds of English words and is very versatile and productive (acidic, prolific). -ikos is just one of several possible adjectival suffixes English has borrowed from Greek and Latin. Another example is -ian (Latin *-iānus), like "Olympian" or "Canadian" or "Trumpian." We tend to prefer certain suffixes in certain linguistic contexts. "Trumpian" sounds right to an English speaker, but a "Trumpic tweet" sounds off.