In general, there probably wouldn't be a criminal or civil case for someone who gets an account at the age of 12 (13 is generally the minimum age) or if someone gets access to an NSFW site at the age of 17 (18 is generally the minimum age) or if someone joins a dating site for adults at age 17 by inputting an incorrect birthdate. In most cases, I would assume the site would just block the user from the site after discovering this.
However, pretend that I was super famous (or the child of someone famous) and the news discovered that I had done one of the above and reported on it. Would I have committed a crime or a tort or neither? Or is it ambiguous because of unclear legal precedent? Are the laws limiting underage access to sites (below 13 for some, below 18 for other) targeted to the site owners or users? If it is a tort, then what damages can the company sue for, since an additional user on the site would not seem to cause damages, aside from possibly negative press (which would be difficult to quantify)? If it is a crime, then what crime is it and what is the statue of limitations?
Does it make any difference if the person was underage at the point of account creation, but no longer underage?
This assumes the person is at the age of criminal liability.
P.S. I know this is similar to Who is responsible for ensuring that 18+ content is not shown to minors: the site or the user? and could be considered a duplicate, but this question is slightly different.
Also, how is this legally different from drinking or gambling underage (for example, suppose I go to a college party and am given a drink)?