In the US, you cannot say that you are an attorney if you are not actually an attorney. Doing so is a crime: fraud. This is important, because if you're paying money to someone to be your attorney, it's important that they actually know what they're doing :)
In the US, to be an attorney, one needs to pass the state's bar in the relevant area of law that one would practice.
> In the US, at least, are people not allowed to represent themselves as their own counsel?
Sure they can. But they can't say "I'm an attorney." That's what OP is getting at.
For more context -- In the US, anyone can sue anyone for any reason. However, that doesn't mean it's going to be a valid or even successful suit. Except in small claims court, representing yourself in a legal situation is an absolutely terrible idea. Not only will you most certainly loose the suit for not following the proper legal procedures (pesky things like having an actual legal argument, presenting valid evidence, citing relevant laws and case precedents, understanding the opposing counsel's legal argument, evidence, citations, etc.) you could easily get yourself into a situation where you'd be wasting so much time and resources by fucking up that you could be slapped with a fine yourself! Or, let's say you're in court, have no idea what you're doing, and get so frustrated you start becoming unreasonable -- contempt of court, put into jail until you calm down.
tl;dr You can't say you're an attorney if you're not one, that's a crime (fraud). If you interact with the legal system without an attorney, you're going to get screwed. :)