A bunch of interesting ideas and opinions here, but the important question IMHO is not between Civ Switching and Continuous Civ but in how to make the gamer's decisions as impactful and meaningful as possible.
Which, I think, demands that the game allow the decision to switch Civs at various points, or to keep on with the starting Civ regardless of consequences.
And to choose and accept various changes to your Civ, either the old or the new, based on in-game events and their consequences - and the gamer's choice of strategy and the type of game the gamer wants to play.
Specific Example: Gamer wants to play a Conquering Host game, starting as Rome. Switching to Mongolia in Exploration would seem like a good choice given the gamer preference, but the option should also exist to remain as Rome with a more cavalry-heavy army (the Equites Sarmatii option for Rome?) OR become Byzantium, with modified 'Roman' characteristics and a cavalry-heavy Tagma/cataphract army as a Byzantine UU.
The more choices, the better. The more Important to the play of the game those choices are, Even better.
So, in the above example, Mongolia, Rome and Byzantium, even though the latter two are distinctly related culturally, should be very distinct and different in their basic characteristics based on their Real Life models, but the game should also allow some very distinct and powerful modifications of those Civs based on the gamer's in-game situation. A set of +1 benefits just won't do: a modified Mongolia should both be recognizable as Mongolia and also be and play very differently from Rome - or even the cavalry-heavy Roman-related Byzantium.
I think part of the set of mechanics needed to help with both immersion and identification is to divorce most of the architecture from direct relationship to the Civ. That is, there should not be a Roman style of buildings, but rather a Mediterranean style that is the basic style for Rome, Greece, or any other Civ based originally in the Mediterranean basin. Civ has done this in the past: related architectural styles to the regions/biomes rather than the Civs. Unique Buildings, of course, should remain just that: Unique to their original Civ.
So, IF you remain as Rome into Exploration, your cities will modify their looks to an Exploration Mediterranean style, but if you Civ Switch to, say, Shawnee (because you've decided to play a Trade/Diplomatic Rome in Exploration) then you should have the choice of Mediterranean Exploration architecture OR American Native Exploration architecture - with Roman Unique structures still around,
In fact, I would make the choice of architectural styles always an option: if you want to play Antiquity Rome with East Asian settlements, that should be your choice - possibly because you have decided that Roman civics and culture more closely resemble the Chinese Western Zhou Dynasty you want to play instead of the game-given Han: your choice, always your choice as the gamer regardless how bizarre (Full Disclosure: I have played Civ VII Byzantium of a sorts simply by renaming Mayan settlements and using Charlemagne's free cavalry to build a Byzantish army: it ain't perfect, but it worked for me).
By giving the gamer the choice of Civ, architecture, Leader combination, and free (within in-game situational constraints) choice of continuing Civ, architecture, and individual bonuses, I think we also give the gamer the maximum possibility of 'telling the narrative' the way he/she wants to. It also gives the gamer the maximum possibility of developing a set of choices that maximize the immersion/identification for that particular gamer.
Improving the game, of course, will require more choices of Civs and Leaders, and for some gamers probably more distinctly associating Leaders with Civs by some kind of Bonus/modified Memento system.
Civ VII already requires Civ choices for each Age. Having set up that system, the game also potentialy allows the choice of Not switching Civs, and potentially of keeping the same Civ or a different Civ but modifying them both by play potential and visual appearance (architecture) as the gamer desires.
