I feel like this comes down to a miscommunication.
Not that this miscommunication is anyone's fault, mind. It seems to stem (potentially, at least) from either differing definitions of "best friend" or from your friend feeling they are closer to you than they feel you are.
Start by understanding why this bothers you. If it's just about putting words in your mouth, that's certainly something to discuss, and a much easier topic (in my opinion) to address:
Joe, I've noticed you speaking for me on certain occasions, saying things like X or Z. (Add in any other times of speaking for you that you can think of, because that detracts from the best friend thing.) I would like you to please stop doing this, as it makes it sound like I've said things that I haven't, and one day it could upset someone and start a conflict that no one wants.
If this is not why this bothers you, and it truly is the fact that they assert they are your best friend, then consider the steps below.
Disclaimer: The remainder of this answer will not work for everyone. This answer may still result in you losing a friend over this matter. This is a very tricky topic to discuss without hurting feelings. Consider your friend's personality before using this approach. And consider how worth the risk of losing this friend it is to correct this behavior before taking any steps to do so. Of all possible solutions I can think of, most have a high chance of upsetting or hurting the other person.
Sorry for the use of bold, but I find this incredibly important to stress. This topic may well end in disaster no matter how you handle it.
If this is the case, and you won't know until a conversation is had, then a discussion on your definition versus theirs may be in order. How do you define "best friend"? Is it your oldest friend? The friend you spend the most time with? The friend who has had your back the most often? Your first step should be understanding your own feelings on the matter.
This definition can be as loose or as strict as you like. This may even lead to you reevaluating this whole situation, though it sounds like it will not. It sounds to me like you already have some sort of picture of your definition of "best friend" in your mind.
Once you've sorted out your definition, find a time when you and this friend are alone. You do not want to make this a public discussion. That will only guarantee hurt for your friend, and make you look like a jerk. (I assume you are aware of this, but other readers may not be.)
Start this discussion by talking about them and how they feel about "best friends", such as by asking
Hey, Joe, I've been wondering lately... What does "best friend" mean to you? How do you define who is your best friend?
This opens the door to the possibility that they have the same definition. If they don't, continue by explaining your definition. (Assumptions about your definition are made here, so sub as needed.)
I hadn't ever thought of it that way- To me, a best friend has always been the person who has been there for me the longest and through the worst of things. Unfortunately, by my definition and how I feel about things, it's not right to have more than one best friend.
This starts to pin the blame on yourself. Remember, likely no one is at fault, but you don't want you friend to feel accused of being to blame.
From here, they may understand immediately where things are going. They may not. If they do and wish to stop discussing things, drop it. Do not push them for a conversation they are not emotionally prepared for. If they don't, you may need to explain further, such as
You are a great friend, because X, Y, and Z. But Rob has been my best friend by my definition for years, and you're not quite there yet. I feel uncomfortable when I hear you tell people you're my best friend- You are one of my great friends, and I have (only a few, many, a handful) of those, but you are not my best friend by my definition.
As you feel necessary, add how you think this would make them feel, or apologize for telling them this.
They may take this poorly. Not everyone can handle this conversation. See the disclaimer above one more time. I don't advise this for emotionally clingy people, or for people who take things more personally than they need to on a regular basis. Think carefully before having this conversation. Once more, read the disclaimer again. Seriously. I want anyone considering any of these answers, as great as they may be, to carefully consider how important this issue is to them versus the risk of losing this friend.
Feeling closer than they are
This, likewise, is a tricky topic. Your definition question should still be the start of this discussion, to rule that approach out. If they have the same, or incredibly similar, definition of "best friend" as yours, consider the following.
We're on the same page, then! That's how I feel about best friends, too.
This could be a point to say something along the lines of, "This is why we're such great friends, we think alike so often" or some such, if this is the case. You want to reinforce that you find them a good friend or a great friend, depending.
Then, you will likely want to follow with a similar vein as the end of the previous approach. Heck, it may even be the same ending. But try to turn it so it cannot be taken as their fault that they feel differently about the title, for lack of a better term, than you do.
Again, this may not end well. Again, re-read the disclaimer. Have you read it again? Read it once more. Memorize its message. This is a topic that many people can and may take incredibly poorly.
I've used a lot more bold and italics than I like to in this post, but I want to make it crystal clear that none of these answers can possible guarantee that your friend will 1) change their behavior or 2) take this without getting upset.
I'll end with this, because I really feel I cannot stress this enough: No matter how you word this conversation, and no matter how close you two are as friends, you may lose them over this conversation. Is this behavior really worth it to you to lose this friend?