To be honest, I think May is a hard negotiator. I don't like that we're leaving, and I voted Remain, but I think some of this is about setting a hard face to the negotiations.
Before I say this, I don't think this is what is morally right, but I do think this is what is happening.
The things like finding out who is a non-UK worker, which children are British born etc, it's about exploiting the one chip Britain has got left with some European countries- their people who are already here.
A lot of European countries have had pretty stagnant job markets for a long time (even pre-financial crisis), and then the number of people who moved in the aftermath of the crisis is large. France, Spain, Poland would struggle to find work for all the people who are currently here if they had to return home suddenly.
Equally, Britain was I think the only country that didn't press the 3 year pause button on free movement of people when the Eastern European countries joined. With the consequence that a lot of Eastern European people moved to the U.K., especially initially, as it was one of the few options open to them. Then later, their communities here where already established so it was a more attractive option than the other options.
So, there a lot of other European countries who arguably didn't want a large scale, quick influx, that's why they pressed pause in the first place when the Eastern European countries joined (and it could be that they didn't want a large influx from anywhere, not anything specific to it being Eastern Euterpe or religious). But they might face it now, if suddenly a lot of people who had moved to the UK have to leave, and don't want to go back home (for the same reasons that made them leave in the first place).
And that's the bargaining chip that's left. (Some) access to the free market or your people have to go home/to another European country.
Admittedly, not many Germans here, so "you have to take your people back" is not much of a bargaining tool with the Germans. But they might not want a large scale influx from other countries, as they are struggling already to integrate the refugees they have taken. And as a (perceived) wealthy country with lots of work they would be high up on the list of alternative places to go if you had to leave the U.K. fairly fast and didn't want to go home because there is no work there/fewer opportunities. Add tariffs on luxury cars and there is starting to be a case for limited access to the single market for the U.K.
I think it's ugly, but I also think that that is what is happening.