We have a civil service - this is their job, their responsibilty, and their mandate.
The haste with which Cummings (who has refused to answer questions from parliamentarians about his role in the Vote Leave criminal enterprise) demands this data suggests that this is a cynical grab for data which may be useful in an election or to target ads at Brexit supporting users / businesses.
This is politicisation of civil service data. The Conservative party does not own this data, and should not and must not receive it.
My guess would be it's "please make sure Google Analytics data for all government website's is shared with us".
The article suggests all kinds of Orwellian tracking going on, but my guess is they're simply trying to get stats on how many people are struggling to register for various government services.
That would suggest that it's unlikely to be anonymised data collection; at best perhaps pseudonymous... and then you get into the questions about how easy it is relate these ids to real identities. Let's hope Facebook stays out of the equation with their fr and c_user cookie fields.
The intentions may be well meaning (let's optimise services for businesses and people prior to significant changes).
However, we've recently had the Windrush scandal and Home Office hostile environment policy: there are over a million people in the UK who may have questions about their residence rights changing and wouldn't like to be tracked whilst they try to find answers.