Can a state space model have changing state size over time?
Short answer: Yes but no. 

Long answer:

If a system matrix is rectangular then it means either; 


There are more number of states than the number of their derivatives which is not meaningful. If you have a system such as a jump linear system or something that couples to the initial system and extends the state space, you can still work with interconnections of square systems that are switched on/off (say, event-based) without resorting back to this esoteric method


or 


There are more differential equations than the number of states which imply that some of those states can be considered as exogenous inputs which in turn can be reconciled by taking those terms to your B matrix and leaving A matrix square again. 


So in short, with square state matrix you don't have any loss of generality.