It is transferred from city to city. If you ask someone how many World Series the Dodgers have won, you don't get two answers based on their time in LA and in Brooklyn. Same with the Giants in NY and SF. The Lakers don't only include their Championships in LA, but also in Minneapolis. There is a reason that the Jerseys of Oscar Robertson, Sam Lacey, Tiny Archibald and Jerry Lucas are in the rafters in Sacramento. They are all part of the same franchise in the eyes of the team and the league.
While what you say is generally true amongst fans and media, I don't believe it's true for all. Might not even be a majority.
I mean, as a KINGS fan since they moved to SAC in 1985 I never talk about or claim the 1951 Championship won while playing in Rochester. Even though the banner is rightfully hanging from the rafters.
As a life long Dodgers fan, I only consider their accomplishments after the team moved from Brooklyn to LA in 1958. The rationale being that had they never relocated from Brooklyn, I wouldn't have ever become a fan.
I've made that same point to my rival Giants friends. They like to cite and claim titles won in the 1880's when the team was not only still in NY but when there were only 8 teams in the National League and the American League hadn't even been formed yet. To me that's laughable.
Makes much more sense to debate the accomplishments of the SF Giants and LA Dodgers, who both relocated in 1958. And if neither had done so, we wouldn't be having spirited arguments about them!
I get that, statistically and historically, all the accomplishments travel with the franchise. But as a fan I don't claim or tout any of it.
To a smaller degree I have a hard time touting the Miami Dolphins perfect season in 1972 or their 2 SB wins. Because I was too young to remember any of it! Only difference is they were always in MIA and never moved. So I acknowledge it but certainly didn't get to experience or enjoy it.
Same goes with the LA Dodgers titles in 59, 63 and 65. But when arguing with my friends, it's a much more apples-to-apples comparison between the Giants and Dodgers because of when they relocated and also when both leagues were in existence.
Just my view on it.
Straying further off topic, it's been common practice among media and the MLB itself to cite stats and accomplishments from 1920 on, after the deadball era.
In the NFL 1970 is often used a benchmark since that's when the AFL-NFL merger took place. Anything that took place prior to the 1st SB season in 1966-67 is largely swept under the carpet.
The NBA doesn't seem to have a delineated time frame, other than perhaps the NBA/ABA merger in 1976.
In tennis, they pretty much only talk about records achieved from 1968 on, which is considered the beginning of the "Open Era".