The problem with this change in law is defining the term 'cohabitee'. AFAIK (and someone with more legal knowledge may correct me on this!) there is no one binding 'definition' of this currently in force and it will be years before the legal system can create one. Until then, any two people living in the same accomodation could technically be co-habiting, because that is what they are doing and that is what the term actually means.
The issue is: how do you separate those people who are actually a 'couple' cohabiting from those people who simply live together? And make it provable in court, beyond a person's say so. The status of marriage clears this up nicely, but I agree it has religious overtones (although my marriage was civil only with no religious content at all)and can see why some people have issues with it.
The solution, perhaps, is for the goverment to get rid of Civil Marriages and institute Civil Partnerships across the board, reserving the term Marriage for those people who want a religious definition. This could then allow those who don't to simplify the declaratory process into a simple ten-minute office visit where you just both sign a piece of paper to say you are partners and wish to be treated that way under the law. (This would also have the advantage of removing the gay/straight divide in such issues once and for all, but that's another argument)
But unless there's going to be years of test cases and chaos and attempted fraud and what have you, I think there does have to be some form of formal declaration made. And I'm not sure I understand the issue with making that declaration. I can and do understand not 'marrying' but does the same resistance crop up in terms of having to state, on paper, simply 'We are a couple. We have children. We live together. I would like this person to be my next of Kin.'?
Something like this would resolve complications in the case of a medical emergency as well, btw. If you're not 'married', currently, and your partner does not or is not in a condition to state that you are, then, legally, you are not his/her next of kin. It's unlikely to happen, I know, but if it ever came to it, the law sides with his/her mother/father or adult children over a partner with whom there is no formal relationship. One for those with MIL issues to consider, maybe...
I know you could simply make a Will, but that wouldn't help in the above emergency and, frankly, Wills can be and are challengable in a way that 'marriage' and civil partnership is not.