As a former assistant, the name thing, I think she's trying to build a rapport and trying to be more familiar OR she's doing the "she let's me call her X because I know her so much better" thing. Only you would know what she's really meaning with it.
Speaking about your personal business could be as simple as her not realizing that other people didn't know about something and when someone asked where you were, she told them, thinking they would know who Y was (she is new you said so that's a pretty easy mistake).
Your structure at work would determine the method of dealing with it. At most of my assistant type jobs the big boss would be the one talking to the assistant. Though those disputes were usually between other assistants etc. The other types of jobs the boss would handle it themselves. As you're not sure what you're supposed to do, I'd ask your big boss. Let them know you've got a couple small issues you'd like to address with your assistant and you wanted to check whether it's something you should involve them in, a third-party witness, or if you are supposed to handle it yourself.
Either way, I'd address it like this:
"X, I thought we could have a little debrief now that you've been my assistant for weeks and see how it's all going. You're doing really well at and ___ so thank you. How are you finding things? [talk about her issues if any].
I do have 2 things I need to mention. First is I must insist you call me [last name]. I know you think of it as my "last name" but I don't think of it that way. In my culture it is more common to be called by a last name and, unintentionally, I find it very jarring to be called [first name] so I would appreciate it if you could remember. This is also important when meeting clients/customers. Please remember to ask them how they would like to be greeted.
The other thing is it's come to light that you've told some people about my personal business such as about ___. Can you explain how that came up in conversation? [depending on her response] I'm sure you're aware that having access to my personal calendar means that you're aware of things that other people are not. I do expect you to keep these things to yourself. Confidentiality is extremely important in this line of work (I'm assuming and projecting here), both with clients and also of course personal business.