Yes and no. The answer to this part is "yes":
That is, when they are in the dining table for having breakfast, it would be one of a couple of options to have--What would you like to have?. When they are in Over the Shoulder Boulder Holders, a bra shop in J. K. Rowling’s novel, it would be ‘What would you like to get?’ In a cafe, the previous bra-shop’s father-in-law’s, it would be ‘What would you like to drink?’
But I think the answer to this part is "no":
In the sentence above, is there some implication that is not said, which definitely be shared between by the speaker and hearer?
The possible options are definitely shaped by the situation, but this isn't really an extra "implication", it's just a background fact. The same thing is true of almost any sentence. If I ask "Where's David?", you need background information to determine who "David" is, and you also need background information to determine acceptable values of "where". (If we're in the U.S., then "He's in England" is a possible answer, but "He's in the U.S." probably isn't — unless David is, say, a fictional character in a TV show, in which case it depends on other background information about him.)