For me, this isn't a "concept": this is reality. Coming from and being part of the furry world, I've seen this happen countless times. This isn't uncommon. It's happened to me as well. When it happens, it certainly isn't acting because acting is fake, a non-reality. When this occurs, the person doesn't even realize it until someone points it out.
Typically it starts with the easiest traits to change: speech, gait, and clothing. As it "worsens," body language changes and core beliefs begin to be molded. It just progresses from there. You can always tell when someone is trying to take on their character: those trying are clearly acting, their actions are not fluid, their conversations are often stilted with delayed responses, and they often appear uncomfortable in their own skin. Slips often occur in those who are trying to be their beloved character. When this happens for real, it's very subtle at first and it's an evolutionary change that can be permanent or temporary.
In terms of personal experience, it's happened in varying degrees. My clothing has changed (permanently) and some parts of my personality have taken on characteristics of characters I've written. These characters shared nothing in common with me. A few events come to mind that are glaring examples of how my personality has changed through roleplay. If one were to compare my core personality now versus, say, high school and show it to someone else, they'd say they were two completely different people.
Another thing I have noticed is this is more prevalent in those who don't like their current lives/situations. Roleplaying is an escape in as much as an exercise. If a person hates their current situation and they fall in love with a character they've made, they may begin to adopt (unconsciously) their character as themselves in an attempt to better their situation. I've witnessed this change a few times. Luckily, it's always been for the better but they almost always end up being an exact duplicate of their character.