I don't think that Brexit is the cause of division in the UK, I believe the divisions were already there to begin with.
The Brexit referendum caused a shock amongst the establishment, they completely missed the mood, it galvanised the UK left to momentumsize themselves into action, they have sought to increase the division, spurred on by the election of Trump and the rise in Antifa (an oxymoron). Threats, intimidation, control of the media, curtailing free speech and violence are now part of the left's agenda, where they have been relatively benign for 20-30 years. In relative terms, there is very little equivalence from the right other than a few protest marches.
Just before the referendum was the opening of the floodgates to mass migration into Europe by the EU, that just poured fuel more fuel on the fire. So the Brexit vote only highlighted divisions that were gently simmering in the UK, probably since the early 90's when Political Correctness was introduced. I would bet that PC provided the spark for a 50/50 split amongst the views of the UK public.
The rise in division has spread to Europe and the States, although I suspect that the same divisions have been simmering amongst those populations because of Globalism spreading PC and corruption amongst major corporations, it's all linked. Other Western countries such as Oz, Canada, NZ following the same path.
Then throw in the rise of the internet and social media and you have pretty much a perfect storm that came into being with the Brexit vote.
Very historically significant in this new Epoch of human existence 😎
What you seem to be suggesting is that EU is being blamed for something for which it was not responsible, that the problem has only been brought to the surface by Brexit.
That would be similar to the case in Spain, where power has been shared by two parties since Franco's death. We, now, have four main ones.