When Thatcher / Reagan neoliberalism came along, it was guided by the concept of the "rational consumer". Individualism was the key to freeing the people from the State. It would, so the theory ran, allow us all to rationally chose the life we want and, by those rational choices, create a fair and prosperous society.
Of course, it hasn't worked out that way. Society isn't equal or fair, and it's only the wealthy who prosper. Yet the concept of the "rational consumer" has become instilled into society. Identity politics is about liberating the individual by protecting them from any and all possible societal restrictions, so that they can choose the life they want to live. It views society as inherently flawed and corrupt. It isn't "leftist" and it isn't even "liberal". It's libertarian, and comes straight from Ayn Rand's playbook.
As with many political ideologies, it's capable of hypocrisy. If a particular movement comes to dominate an area of libertarian society, it strictly enforces its moral code to make sure people are libertarians in the right way. So, religious right-wing libertarians will oppose abortion because pro-life is the wrong kind of personal choice. Left-wing "progressives" oppose laws that protect women because giving women autonomy through protected spaces and sex-based positive discrimination is the wrong kind of personal freedom.
I think our current political environment is best described as ideologues vs realists. The ideologues are the Brexiters and the Corbynistas, whilst the realists are the centrists and left or right of centre liberals. Whilst we expect libertarianism on the right, I think we've been caught off guard by libertarian entryism on the left. By donning the terms "liberal" and "progressive", and by claiming to be for social equality, rather than just personal liberty, the ideologue libertarians have hijacked left-wing politics.