It's all just hokum, Ideally you want the same make, model and size that your car was delivered with, that's what the car was developed with, anything else will give a slightly different driving experience, which may be better or worse in what is important to you.
Just make make sure you have the same make, model and size on each axle, obviously a different a tread pattern, size, compound or internal structure will give differing levels of grip, breaking efficiency, cornering ability, comfort and noise levels. In extremis, this could lead to an unpredictable driving experience especially when fitted on the same axle.
I don't understand why people put completely different sizes on their car and then state you can't put run flats on the front and normal tyres on the rear, changing make or model or size will alter your cars handling as will having extra load tyres or run flats