"at me" vs "at mine"

I'm watching (an American) TV Show where one of the characters says: "[..] treacherous, despicable serpent. Hurling wicked insults at mine every turn". I instantly noticed how they used an unusual "at mine" instead of "at me". I believe this is the first time I hear someone using that expression, so the question is: is it grammatically correct?

No, this is not standard grammar; however "at me" is not right either. The right form is "at my", where "my" is a modifier of "turn". "At" is not the usual preposition used with "hurl" but the preposition introducing the adverbial of place "at every turn" which takes here a special form: "at my/her/his/ every turn". This idiom is not attested in the dictionaries but is found in plenty.

ngrams: at my every turn, at her every turn, at his every turn

The idiom that explains it and that is found in the dictionaries is "at every turn".

(SOED) at every turn at every change of circumstances, at each new stage, continually

A translation of this in Chinese exists and shows that the correct determiner is "my" (found in this page).

It is correct but doesn't, usually, mean the same as 'at me'. 'Mine' in this case almost certainly means the speaker's family and other relatives. The term is most familiar from the expression me and mine which Lexico defines as:

My family and relatives.

with examples:

I consider it a great compliment to me and mine that you would consider our family worthy to be joined.’

and

Tell your minions to stay away from me and mine

I can't tell from the short context but I expect that the 'Serpent' has been verbally, and possibly physically as well, attacking the speaker's family rather than the speaker personally.