From a couple of posts I've seen it looks like they put up our badge which is probably just something like their web guy being told to add a fixture against Cov and having no idea Coventry United existed. Not sure there's anything more to it than that. As soon as they realised it was changed, not really any need for a statement.
Personally if I had taken over Cov Utd Women, which I wouldn't as it just seems a way to burn money, I'd be approaching CCFC about a rebrand even if the club are putting zero money in. I know this game is away but if you had Coventry City Women playing Arsenal in the cup you'd probably get a few of our fans popping along and given how small their crowds are that's got to be worth considering.
You're spot on. All womens teams that are affiliated to a men's team, form part of their website.
Others like Durham, Lewes are obviously not so are stand alone and therefore don't have the financial support of the overall club.
Someone at Arsenal would've heard Coventry and put 2 and 2 together and got 4.5 but its an easy mistake to make.
I like your point about the approaching CCFC to become part of the family so to speak but I'm not sure they'd take it on, even with no costs as I'm sure eventually the womens team would need some. Also the club would likely want them to do well if they're using the name/badge etc and they won't do that without funding.
It doesn't cost millions and millions to run a womens team (unless you're Chelsea or Man City) but working with Aston Villa WFC, Leicester City WFC and Watford WFC, I know that they can spend as much as Cov men's team on hotel stays. Despite less fixtures and less hotel stays, it still adds up to a sizable amount over a season.