Does this sound rude?

Really? Your new employer's an idiot. They should call you by your first name.

It's that weird kind of rude where people try to be overly accommodating that ends up making them feel uncomfortable. A lot of people don't get it.

I do not think that was rude, if you usually go by your middle name maybe she thought she was being accommodating. I do not see you saying she said no, you cannot go by your first name, just tell her you want to go by your first, I doubt she will care. You are reading way too much into it. Had you have simply answered her question without saying you usually go by your middle name there would have been no confusion. If you are trying to go by your first name now why even bring it up.

What's going on with names lately...my co-worker who's been there just over a month now has changed his name 3 times.

We often project our own insecurities onto others without realizing it. It sounds as if she is insecure and thought you may be as well. Just trying to be polite by using your first name. In her heart she was trying to encourage you saying it's ok if you wish to change your mind.

She might have been thinking you'd prefer your middle name but that policy wouldn't allow it, and reassuring you that you could use it. But the way she worded it was poor. I'd have said that I don't have any problem with you using either of your names and just to let me know which one you chose.

You told her you use your middle name. It sounds like she took that as you wanting to be called by your middle name.

That sounds really strange because most companies have a policy on what to put on a name badge. Even if that policy allows you to choose.

Isn't she giving you the choice to decide?

I think she was trying to accomodate your wishes, since you mentioned that you usually go by your middle name.

Hmm, I don't know.
What's your first name?

You got a black sounding name?