I have to admit I find the concept of “immersion” kind of obscure. I spend a lot of time as a GM, meaning I have to take on the roles of several or even many different people in a session. I have had the experience of spontaneously thinking of actions or lines of dialogue that fit a particular character, without scripting them in advance, not even a few minutes in advance; I just know what that character would say or do. But this doesn’t involve my experiencing the situation as if I were that character (which I agree would be disruptive). It’s third person rather than first, more in the spirit of “Have you tried acting, dear boy?”
And the same thing happens when I’m a player rather than a GM. I play much more in third person than in first. The character is a persona, a mask.
I had one player, in my first Transhuman Space campaign, who was doing something that might have been immersive play. What I experienced it as was his not doing much with his character and not making her comprehensible to the other players or to me. I talked with him about it and I gathered that he was very focused on how she was experiencing the events of the campaign, but that he didn’t think of telling us, or of having her say or do things that revealed her to the audience; he was focused on his own experience of the character’s point of view. I asked him to adjust his portrayal of her to make the rest of us more aware of how she was reacting—to give the audience more purchase on her, I guess.
But I don’t know if that’s what “immersion” is about, or if that was just a peculiarity of this one player.
However, back at the main topic, I don’t think that playing more than one character makes it impossible to have a strong character concept, or deep involvement with the events of a campaign. Having to play two different characters in the same scene, or to alternate rapidly between the two in short scenes, might be a problem; but if an hour or more focuses on one particular character, that’s plenty of time for most players to get involved in the role, at least in my experience. (My players in the campaign where they each had four characters were very much at the characterization/social interaction/roleplaying end of the spectrum; that was the least action/focused on my three campaigns in that cycle.) Which isn’t to say it would work for what you specifically have in mind.