Can it be taken by a plane as a baby’s “free luggage”? As it doesn’t fold at one unit, I’m not sure if airlines accept it…
This is more a question for the airline than to me, like this, I believe, varies from airline to airline and is up to their final consideration.
In general, however, it absolutely should be able to be taken as a baby's piece of luggage as it is the necessary mean of transporting it - and the design of the folding system should not be 'blamed' on the parents to be a two-piece one. The general rule is that the pram or pushchair must be fully collapsible, which, in theory, even a two-piece-folding one is.
Still, as I said, you absolutely need to check this with the airline you're traveling with. If they won't be ok with that, I'd recommend taking a much smaller ultra-compact buggy with you and have it for occasional use.