Prestige of an undergraduate degree is not meaningless, but how much meaning it will have in your life varies wildly based upon your profession and if you pursue additional schooling.
If you become a lawyer, people will ask where you went to law school all the time, and sometimes form opinions about you on that. People tend to ask where you attended undergrad only when discussing college sports or something similar—it comes up now and then, but usually not in ways that have much if anything to do with your career or social prospects.
If you become a doctor, hardly anyone other than other doctors and related people will ever know where you went undergrad unless it comes up when talking about college sports or taking your kids for college visits. Actually, even the medical school itself often does not come up—people ask what sort of doctor you are, not where you went to medical school or where you got your undergrad degree. There are such fewer medical school graduates out there that the achievement is GOING to medical school, not attending one ranked #10 versus #27. Specialization and residencies matter as much or more than medical school, so your undergrad degree gets pushed further and further down the priority list. Very few people will care that your Biology degree came from X versus Y by the time you become a surgeon, for example.