Any problems mixing brand of winter tyres (same make on each axle)?
I would argue that yes there is, because different tyres perform differently.
So, if in certain conditions, a tyre performs in a certain way on/at a given corner/speed/loading, that is fitted to the outer side of the corner, it is going to perform in a given manner which you may become used to.
Now turn the car around so that the other tyre is on the outer side of the corner it won't perform in the exact same way as the first tyre, simply because it is a different tyre with different characteristics. Not necessarily dangerous, but certainly not ideal.
I know this is a totally different scenario, but for similar reasons, we had a warning at work not to mix Dunlop and Michelin tyres on the same axles on our aircraft.
So if we had a wheel cx to do, and removed a Dunlop for instance, the new wheel and tyre assembly going on in its place had to be a Dunlop - we would then even have to double check the tyre on the other side of the axle was actually a Dunlop too - even though technically, the part number, size, running pressure, and everything else between the different brands was identical. We just could not have a Dunlop and Michelin on the same axle.
(This became a pain because if we had a worn Dunlop on one side of the axle, and only a Michelin to replace it with, we then had tox do two wheel changes so that the axle had two Michelins on it

)
It was also damned wasteful as the part-worn tyre coming off the aircraft that was being rejected simply for its brand, was invariably scrapped