Very similar, the only difference is the first question is asking, "How would you like your coffee [right now]?", and the second one is more, "How do you like your coffee [generally when you have coffee, which doesn't seem to be right now]?" Both are really asking: "So, do you like your coffee hot, cold, drip, Americano, or espresso; just black or with cream, half-and-half, milk, soy milk or oat milk; do you take it with no sweetener or with sugar, stevia, honey, or aspertame?"

Both of these questions pertain to what you put in your coffee such as cream, sugar or milk.

Usually both questions mean “Do you want milk and sugar? Or just black coffee?

How would you like your coffee? - It is something a barista(cafe person）would ask you when you order a drink. How do you like your coffee? - this is something more conversational. Like a group of friends can talk about their favorite kinds of coffee to drink like lattes and cold brews.

How would you like your coffee? Is being used when you offer some one a cup of coffee. But how did you like your coffee is being used when ask some about his favorite coffee