It is a construction of two nouns: maziwa (class 6) + y-a (Prefix-connector) + mbuzi Roughly "~a" can usually/often be translated as "of". So it can be both (just not in this case).
"drink" would be "Ninakunywa maziwa ya mbuzi." - I have never heard of -chukua used for "drink", so I'd advise not to use it that way (though you might be understood) :)
"Ninachukua maziwa ya mbuzi" as I am taking the goat's milk (possessive) - As in transportation.
Well, looks like I can answer part of the question. Literally, it's the "transport somewhere" version: https://en.wiktionary.org/wiki/chukua.
We'll have to wait for others as to whether or not the dual meaning of "take" might also exist for "chukua" (either because it long has, by coincidence, or because it got calqued from English into the language).
Can kuchukua mean to choose, like "I choose the goat milk?" In English we might say "I'll take the goat milk" if someone offers a few beverages. Or is it just to transport? I am confused by this sentence!