I don't like such rigid interpretations of the classes. You don't just "unlearn" things like that in real life when your direction changes. For example, I am a trained engineer, but recently I started architecture graduate school, which requires a very different skill set, engineering being technical and architecture being much more about creative problem solving and visual design. With a rigid interpretation of how skills are applied, I'd be about as advanced as a kid in their first year of undergrad, but with the help of previous training I am far more rounded and knowledgeable. I rarely have to apply that engineering knowledge directly, but it doesn't mean that I can't use those strengths to my advantage to hold my own in classes that mostly consist of students with a 4 year degree in architecture. This isn't anything unusual either, about 1/4 to 1/3 of the students in my (ivy league) program have no previous architecture experience at all.
A barbarian knows how to swing a weapon and hit opponents. That skill can't be unlearned. But when you multiclass you aren't both a pure fighter and pure barbarian. You have that ability to rage, but you don't rage as well as a pure barbarian, which has been curbed by your training as a fighter. You have those bonus fighter feats, but you don't have as many fighter feats as a pure fighter, representing less training as a fighter. This doesn't mean your combat experience gained as a barbarian has to be completely forgotten as if you were newly born, those past experiences can still be applied. If you are mostly barbarian with a sprinkling of fighter, you might not have learned the self-discipline to not rage-out at inappropriate times. On the other hand, if you are mostly fighter with a little bit of barbarian, you might have the discipline to control yourself, but in desperate times or on occasions when you need to overpower your opponent as quickly as possible, that ability to rage might emerge. This is also indicated with the development of your rage ability. If you have 1 or 2 barbarian levels, you can only rage once per day for a short time, so it is best saved for desperate measures, but with more barbarian levels you have more rages per day, indicating that you are more likely to rage out in combat. Also, your fighter training is indicated by your feats. One with more fighter will have more feats and therefore more trained weapon and combat skills. This doesn't make the character any better or worse, only one with different skills and strengths.