a) How would you like your coffee?
b) How do you like your coffee?
The grammar is the same for both of your examples, and they are ambiguous about whether "how" is about "liking" or the coffee itself, and they differ in mode and tense. We resolve it by context.
As an offer
Assuming you've just arrived and we're sitting in my kitchen and socially it would be the moment for me to offer you something, then these two questions are the same: I am offering to make you a coffee, and asking your preference. The expected answer is something like "plain black", "with milk", "with sugar" etc. A is phrased about a hypothetical future coffee, B is phrased about your habit. Because this situation is so common, this is almost a stock phrase. As an offer, "how do you like X" is used when X is something I expect or know is your habit.
After trying it
If however I've just given you the coffee and you've just tried it, B is asking how you like it, and the typical answer would be "Very much".
Question A would be a very strange question which I would only ask if I understood that you didn't like it the way I've given it to you: the expected response could be something like "Could I have one without milk?".
Similar sentences with "how" being about "like"
These only make sense if they are before I've given you the thing.
- "How would you a coffee?" Answer: "very much", "no thank you" etc. (a coffee = a cup of coffee)
- "How would you like some coffee?" Answer: "very much", "no thank you", etc (some coffee = an amount of uncountable liquid)
- "How would you like £100?": Answer "Very much"
- "Would you like to go to the moon?" (Answers: "Yes", "No", "Yes, very much", etc)
- "How do you like the idea of going to the moon?" (Answers: "very much", "Not at all", etc)
Note this last one is about an idea which you currently hold, so I can ask in present tense how you like it.
Similar sentences with "how" being about the noun
In some contexts, it's clear we're actually asking about the noun.
- Bank Teller: "How would you like the money": Answer: "In £20 notes please"
- "How would you like to go to the moon?" Comedy answer: "I would rather go to Mars" (This answer forces a reinterpretation of "how" as being about the destination, not the liking.)
Similar sentences with "how" being about the verb
- "How would you like to go to the moon?" Comedy answer: "By rocket, please" (This answer forces a reinterpretation of "how" as being about the method, not the liking.)
If you are looking at a object, then "how do you like" is about the object in question
- "How do you like your new car?" Answers: "It's great", "Not as good as the old one", etc