It's been a bit difficult for me to find good information about the etymology of the derivational suffix -aster. De Vaan doesn't seem to talk about it. A number of sources indicate that it is from Greek; e.g. Wiktionary says
From suffix originally forming Ancient Greek nouns from verbs ending in -άζειν (-ázein).
and says that -astrum is an alternative form, which seems to imply that it would go back to a Greek ending -αστρον or -αστρος.
But it's been hard for me to find information about the use of -αστρον in Greek. Searching Perseus only turned up 20 examples of words ending in -αστρον, and most of them don't look very similar in meaning to Latin forms suffixed with -aster. The only one that looks close to me is λαίμαστρον, which LSJ defines as "greedy beast', term of abuse": it seems to be derived from a verb λαιμάσσω. The results for -αστρος don't look much better: 14 nouns, of which 5 seem to be derived from ἄστρον "star" and 4 seem to be derived from γαστήρ "belly". I also looked at words ending in -αστήρ (which I'm not sure is even very plausible as a candidate source of the Latin second-declension form -aster) and they mostly don't look too similar to Latin -aster words either.
What is the exact hypothesis about how Latin is supposed to have developed a suffix -aster based on some Greek ending? Or if there are other hypotheses that are currently considered plausible, please tell me about them.
I am aware of one alternative proposal (at least for the word patraster) that was criticized by Fr. Stolz in 1905: H. Ehrlich apparently tried to derive patraster (which he apparently supposed to have a long vowel in the penult: patrāster) from patrāvester, but Stolz argues that the change āvĕ > ā is implausible in this context ("Über Angeblichen Wandel Von Lat. āvĕ Zu ā." Zeitschrift für Vergleichende Sprachforschung auf dem Gebiete der Indogermanischen Sprachen 38, no. 3).
Here are examples of the three main usages that I know of for this suffix.
-aster, -astra, -astrum is found at the end of some adjectives with the sense "somewhat", such as surdaster "somewhat deaf" from surdus. I've seen it called a diminutive suffix in this context (see e.g. this question, the answer and comments: Translating -ish and -aster endings).
It also appeared in the kinship terms patraster and filiaster. The meaning seems to be not entirely straightforward to translate into English, but it appears that these words could refer to either what in English would be called "in-law" relationships or what in English would be called "step" relationships. Wiktionary says that in Italian, "fratellastro" can mean "half brother". I suppose that in theory, these words could have a different origin, but the meaning does seem relatively similar (a "patraster" is somewhat like a "pater", but somewhat different).
It appeared in some pejorative derived nouns like philosophaster "a bad philosopher".