I literally just made my account after seeing this thread and having to add my two cents.
I am friends with all of my exes, and all except one was a mutual break up. I loved and cared for them deeply, and to some degree I still do care about them. Most of them I do not talk to regularly, except the last and most recent.
If a new partner told me I had to cut off these people, who were important to me, important to my growth as a human being, to the portrait of my history- it would be a deal breaker. I have always been MINE before I was yours.
I agree with Slant, this behavior comes off as controlling, if not a wee bit creepy. None of my exes had a problem with my friendships, though one would tease me about it. If my future partner told me I had to cut off these exes, I would definitely not, I'd have to end the relationship. I am not his property or his child who he can tell what to do and who to communicate with. I am fine with him being friends with his exes, so he should be fine with me being friends with mine. Like I said, it would be a deal breaker and I would see this as a redflag for future controlling and possibly gaslighting, manipulative behavior.
There is a submissive, co-dependent type of woman you are looking for that would suit you fine but it's clear this Emily isn't it, she has different values and ideas of boundaries than you do and this WILL become an issue, even if you put it off or try your best to ignore it. You are incompatible. You can try but it's not good for you, you should find someone compatible and get out before you end up hurt. Goodluck!
(And her ex is being obviously manipulative and sounds emotionally unstable to the highest degree. But that is not every case! Not all of our exes are demented and clingy! She's going to have to make a stand for her own well-being before anyone elses. But this isnt a regular case with a regular ex.)