rented impreza .....
I had also driven one before (rental from Downtown ....) and believe me, I cannot wait to get my own car back and I was driving the impreza more like a "bullock cart"
car start to get a little shaky - as in strong vibrations felt in the steering wheel?
which tyres are on which axle?
irregardless of the above, the strongest possibility is that the balancing on one (or more) of the tyres is off
I think the reason for the advice against putting tyres of different brand, construction and tread patterns has got to do with braking and handling especially in the wet where grip, braking characteristics and water dispersion properties are of paramount importance