oh does my heart go out to you. I have a 3 yr old daughter who is any number of princesses during the day, including jasmine.
I am going to guess that she probably doesn't really understand the gravity of what she says when she said that she was killed by Jafar and is now Jasmine. My daughter tho a year younger says stuff kinda like that, not the same involving her being killed, but sometimes her brother (11mo) or her parents or something and so when she says that I explain to her what it would really mean if we were killed, you would never see us again etc... and how sad that would be. So I will often tell her that it's not nice and not appropiate to play like that and have mommy and daddy killed or her even and she normally doesn't do it again.
Perhaps you can sit down with her especially since she is four, explain that you think she is beautiful and fun just like a princess but you are her parents and God gave you the responsibility of naming her and raising her, so she is allowed to pretend to be which ever princess she wants but she is still required to respond to you as the mother when you call her real name.
We sort of had an issue with this as well... she wanted to be Belle, (or who ever, it changed 100 times a day) and would say no mommy I am Belle, don't call me Ellie. So I would but would obviously forget and so I would call her Ellie again and she would remind me again, that got pretty old, especially because it felt like she was telling me what to do all the time and that doesn't last long in our house....
So we did sit her down told her she can pretend to be a princess and be all different ones but her real name really is Ellie and sometimes mommy and daddy will forget to call you your princess name and sometimes we will choose to call you your real name, either way you MUST listen and respond.
So I know what you are going through, for me it's extra hard as I hated anything super girly especially princesses growing up and I swore even 5 years ago if I ever had a daughter there was NO way she would ever be allowed to play with princess type stuff.... but I have grown and realized she is who the Lord created her to be and that is 100% girly girl and I can't stifle that. BUT it does get really annoying that is for sure.
Like you said you and your husband are not living together, so yes that may be a coping mechanism, she is old enough where you may be able to talk to her about the possible connection... perhaps not, I don't really know.
I hope that helped some !!
Edit: I also wanted to say that she will also name my husband and I along with her brother different characters from the stories and that gets pretty old to...I try to play along but some days I just tell her I am mommy and that is it... no Drizella, no Jafar, no abu, no fairy nothing :-)