I don't find it strange at all, if you constantly and consistently interact with the client, the client should know your name, and referring to you by "she" instead of "D." all the time is not a big deal, the client should know who you are. Certainly not something to get upset over. If your boss were saying "someone will help you" referring to you obviously, rather than "she can help you" or "D. can help you" perhaps I might be a little irked, but it's still not a reason to become upset, quit, or complain, unless there is other treatment from her that bothers you and needs to be addressed.
If she says "she is not always at the front but someone else could be sitting there" it might be accurate. I'm sure you step away to use the restroom every now and then, or have a backup person to sit at the front while you use the restroom or go to lunch. Yes, you and your boss are the only ones who handle clients, but if a client calls or shows up and you're at lunch or the bathroom, obviously someone else is handling them during that time, so why would it be inappropriate to say that someone else could be sitting there? Seems logical, and true.
My boss sometimes says "my assistant will send that over to you" versus "N. will send that over to you", and I don't get upset, it doesn't bother me in the slightest. I have a signature block in my email that informs clients of who I am (though pretty much every client knows me and has interacted with me in the past). If he said "someone will send that over" I might be confused as to who that "someone" is and might make a joke about it, considering I am the only person in the office he works with directly. Sometimes, while on the phone with a client, he says "my assistant just walked into my office" (rather than say "N. just walked into my office") and still, the client says "hey N., long time no speak!" and that's that.