MOT Fail - Tyres and their brands

So why have I read it is illegal to have different tyres across the same axle?

You can't have different tyre types such as cross ply and radial mixed across an axle. But you can have different manufactures producing the exact same size and specification of tyres and mixing them is fine.

No concern for mot purposes but i have noticed quite an increase in tyres marked specifically for a particular brand of vehicle such as MO for Merc, AO for Audi, AR fpr Alfa, Vol for Volvo, T0, T1 and T2 for Tesla. Evidently designed to cope with specific requirements for that particular vehicle.
I had a Mazda MX5 that when turning on full lock the front wheels would judder across the tarmac, turned out it should have had specific tyre fitted due to LSD fitted, vehicle was fine after replacing both rear tyres.

As has been mentioned, 'type' refers to construction type.

The three construction types are Radial, Cross Ply and Bias Belted, these shouldn't be mixed across an axle.

In reality, you'll be hard pushed to find anything except a radial tyre. Runflat tyres are radial, even though their structure (sidewalls) are considerably stiffer, and although mixing runflat and 'standard' radial tyres isn't a good idea, it still isn't illegal nor an MOT failure.   

Nope, tread pattern has nothing to do with construction type. KwikFit have done nothing wrong