The genitive is used with verbs of sharing.
πάντες μετεῖχον τῆς ἑορτῆς
all took part in the festival A. 5.3.9, μετεδίδοσαν ἀλλήλοις ὧν (= τούτων ἃ) εἶχον ἕκαστοι they shared with each other what each had 4. 5. 6, τὸ ἀνθρώπινον γένος μετείληφεν ἀθανασίᾱς
the human race has received a portion of immortality L. 721b, σῑ́του κοινωνεῖν
to take a share of food M. 2.6.22, δικαιοσύνης οὐδὲν ῡ̔μῖν προσήκει
you have no concern in righteous dealing H. 2.4.40, πολῑτείᾱ, ἐν ᾗ πένησιν οὐ μέτεστιν ἀρχῆς
a form of government in which the poor have no part in the management of affairs R. 550c. So with μεταλαγχάνειν get a share (along with somebody else), συναίρεσθαι and κοινοῦσθαι take part in, μεταιτεῖν and μεταποιεῖσθαι demand a share in.