Does it sound normal or overly polite to use "Would you like" between friends, for example "Hey Bob, would you like to play chess with me?"
Not necessarily. The difference in formality between would you like and do you want stems from the difference in timeframe. Do you want is immediate, refers to right now; would you like is unspecific, may refer to right now or one day. This is typical of English culture, in which formality is expressed through vagueness, allowing the interlocutor to respond without seeming blunt or dismissive of an offer in good heart, rather than via structure and deference, as in continental European cultures. This cultural difference is reflected in the language.