Usually.

I’ll start by saying that I’m no lawyer, so I can’t provide anything that might be considered legal advice. If you’re really in a bind over something related to this, contact an attorney.
However, I have opinions.
False identities have their place. But while they reduce some forms of risk, they introduce others.

Using your real name
Use your real name and birthday unless you have a strong reason not to. Fake details don’t add much privacy, since most accounts can be tied back to you anyway. Worse, they can make it impossible to prove an account is yours when you need it back.
In general, the benefit is negligible
I rarely use fake IDs and generally don’t see a huge benefit in doing so. For any account used for anything meaningful, there are many ways to tie it back to my real name. It’s just more hassle and something else to remember for the accounts I use.
That Amazon knows I’m Leo Notenboom seems inevitable. I could use a fake name and birthday, I suppose, but to what end? It might make sense to use a dedicated email address1, but it’s an email address that you’d typically want to associate with yourself anyway.
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Anonymity
There have been cases where, on a discussion forum, for example, I chose to present as some identity that’s not directly associated with my real ID.
Many years ago, my wife ran a collectible doll shop. Sometimes it was appropriate to be identified with the doll shop. Other times, it made sense to take part in something without being tied to the business. In cases like that, it made sense to set up a fake ID. I didn’t expect it to be anonymous in any absolute sense, but it hid who I was well enough.
I also maintain a separate ID for online gaming, mostly to minimize the overlap between that and my business.
None of these cases are particularly security-related. They’re lightly privacy-related, and even then, if the barriers between my fake and real IDs come down, it won’t be a great loss.
True anonymity is hard, depending on what you believe the threat to be. Staying anonymous from random strangers in a discussion forum is one thing, but if you’re at risk from government agencies or corporate entities, things get complicated. Anonymity takes more than using a fake name and birthday. It can require completely detached email address(es), careful network use, and even dedicated hardware to keep your real identity separate.
Practical risks
You raise an interesting question, though. If you use a fake name or birthday for these less-important accounts, you are increasing the difficulty of account recovery should you ever need it.
Account recovery can be difficult even if all the information is correct. Many people lose access to accounts in a variety of ways, even when using their correct, current, and accurate information. Fake information sets another barrier.
In some situations, account recovery can require photo ID. Your fake persona won’t have that2. The account might well be lost.
If you can’t prove you’re the account owner, you won’t get the account back. Maybe that’s okay for an account where you’re presenting as an alternate persona, but it’s something I’d consider before making that choice.
Security questions are different
Just to be clear, it’s still considered good security to use fake answers to security questions. Things like your first pet being “noneofyourbeeswax” and so on don’t misrepresent who you are. They simply verify that the person who established those answers when the account was set up is the same person now confirming that they’re authorized to access the account.
The use of security questions is on the decline. One reason is that the answers are often easy for anyone to find. And that’s the reason making up fake answers that you’ll remember later3 is a valuable technique.
Age verification enters the picture
Some sites require that you somehow verify your age before you can access their content or functionality. If they go beyond asking if you’re over some specific age, they may require some form of official identification.
Again, I’m not a lawyer, but it would seem like presenting fake identification in these situations runs the risk of being considered fraud and thus subject to legal ramifications. At a minimum, discovery could result in account restrictions or even loss.
This is an area that’s changing quite a bit right now, and varies dramatically from one country to the next (and in the US, even between states). Once again, it’s worth understanding the risks you run by providing your real ID and the risks associated with a false one.
Do this
My recommendation is to use your real name and birthday unless you have a strong reason not to. They’re too important a part of identifying yourself as the account owner, should you ever need to. Being somewhat fatalistic, I also think that in general, you’re not increasing your privacy very much by using fake information.
There’s no real harm in doing so, but you may not be gaining the privacy you think you are, at the cost of making it more difficult should you ever need to prove you’re the real account owner.
What’s your experience? Do you have scenarios where fake IDs are valuable that I’ve missed? Let me know in the comments below.
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Footnotes & References
1: Which, coincidentally, I do for Amazon. It was originally to see if they shared my email address. So far, after many years, they have not.
2: I suppose you could use AI, but you run the risk of it being detected and making things even worse. Personally, I’d expect an account to be permanently disabled if an AI-generated ID were provided.
3: Password vaults are a good way to do this. Add the answers as a note for the account.

