As a native English speaker "any milkshake" sounds wrong to me. "Milkshake" is a countable noun (like "cappuccino" and unlike "milk" or "coffee"). "Any milkshakes" would be better, unless you are referring to the milkshake left in your cup. "Is there any milkshake left in the cup?" That somehow works.
The problem with trying to force a translation of the grammatic construct is it changes the meaning. I believe the meaning of this sentence is "Do they have milkshakes" i.e. products I could purchase. "Any milkshake" would really be referring to the liquid itself in a way that I don't think the finnish sentence implies.
The trouble is that the "pirtelöä" partive absolutely refers to the liquid itself as a mass noun, not a countable thing. If you wanted to (clearly) refer to products that can be purchased, you'd use the plural partitive "pirtelöjä" or plural nominative "pirtelöt" depending on the context, and translate both as "milkshakes".
What my comment meant is that "Do they have milkshakes" is the way most English speakers would try to establish if a shop could aid in their quest to put milkshake into their belly. Where as "any milkshake" would require some contrived scenario: "Do you have any milkshake left in that cup? I want to feed these ants."
How would a Finn normally ask a third party if a shop sells milkshakes?
Saying "Onko heillä pirtelöä?" or "Tarjoavatko he pirtelöä? (Do they offer milkshake?) doesn't sound strange at all in Finnish, but "pirtelöä" is a mass noun there and understood as such.
Some example sentences with "pirtelö" (which by the way comes from the word "pirteä"" - lively, perky, alert) when not in partitive/used as a mass noun:
"Haluaisin (suklaa)pirtelön, kiitos." - I'd like a (chocolate) milkshake, please. (one countable thing)
"Pirtelö maistuisi juuri nyt" - lit. A milkshake would taste right now, i.e. it would be just the thing, hit the spot (one countable thing)
"Pirtelöt loppuivat kaupan hyllyltä" - lit. The milkshakes ended from the store shelves, i.e. the store ran out of milkshakes (several countable things)
"Haluaisitteko te (, että ostamme) pirtelöt?" - Would you like (for us to buy) milkshakes (several countable things, each person gets their own milkshake)
It might be a direct translation, but i know no native american english speakers that would ask "...milkshake" with or without the "any." It would be "...milkshakes" with or without the "any." The subtlety with "any" is that using it implies that they normally have them, but they might not have any at the moment. Without the "any" it implies do they normally have milkshakes at all.
As a native English speaker, I'd like to add my thoughts to the pile. "Milkshake" in English is a countable noun, not a term for the substance itself, unless the substance is, say, smeared on your shirt, as in, "you spilled some milkshake on yourself." You would never ask if someone "Has any milkshake." Maybe because milkshakes tend to be made one at a time? For whatever reason, "Do they have any milkshake?" is a pretty awkward sentence.
Agree with most people here regarding "any milkshake", it isn't right in English. I can see how "some milkshake" can sort of work, but if you are using "any", then it has to be "any milkshakes", since it is a countable noun in this case. The best solution grammatically would be "Do they have a milkshake" or "Do they have milkshakes", but not any milkshake.
I totally get it that the course developers are trying to drive home the point that milkshake needs to be partitive in Finnish. However, insisting on an English translation that adheres to that, results in a completely non-English sentence. Maybe milkshake should just be dropped from the vocabulary presented here, since it seems to produce such antagonism!
Here you can read about the rules for this: https://uusikielemme.fi/finnish-grammar/morphology/vowel-harmony-vokaaliharmonia-finnish-grammar/.