This base type goes back to Latin mŭlgēre 'milk' (cf. also grc. ἀμέλγω, where the initial alpha is obviously to be interpreted as a prefix, although its motivation remains unclear for now). The Romance equivalents such as ita. mungere (cf. Treccani, under mungere) or lld. mùje (cf. EWD IV, 488), however, presuppose conjugation change to mŭlgĕre and a shift from -l- to -n-.
A connection with Malga is logical semantically, but phonetically problematic because of the differing sound vowel. According to Kluge, 614 lat. mulgere, like deu. milk and grc. ἀμέλγειν, goes back to ine. *melǵ- 'to milk'.