Normally, no.

You’d have to set up a completely new account which you only ever access via e.g. Tor, then use Tor Browser on iOS to log into that account and only ever use your account exactly like this or else you’ll leak your current IP address as well as related data about you or your device to Meta. Also you’d have to never give any kind of personally-identifiable info to them. Then you’d have an anonymous account, but the goal of Facebook is to connect to other people you know. Once you add and talk to friends on Facebook, they might already know who’s behind that account, especially if you already had a different account in the past which pretty much had the same set of friends already. Also, remember that many people use WhatsApp (also by Meta), and WhatsApp collects among other things the whole address book or contacts list from the phone of each user. And most normal users don’t use contact scopes or things like that to spoof their contacts to Meta, so they’ll most likely get the real contacts list. That means if let’s say you have 5 friends named A, B, C, D, and E, and those 5 friends all use WhatsApp and have each other and you saved as a contact in their address book, then it’s even easier for Meta to guess with high precision that you are this person that knows those 5 people and/or who is known by those 5 people. So there are even other factors to consider, some of which are outside of your own control, because all the persons you know and want to add to Facebook give a lot of data to Facebook themselves as well, and that data also might be used to identify you in various circumstances. So you’d also have to behave anonymously when using your anonymous account. Which kind of goes against the whole thing of social media like Facebook. Since they monitor every user’s interaction on their website or apps, it’s likely that they’ll be able to eventually identify you once they’ve gathered enough data about your usage patterns, visited links on Facebook, contacts you added or which have added you, messages to contacts or from contacts to you, and so on. Simply using Facebook means you’re actively giving tons of data based on your usage alone to Facebook. It’s even theoretically possible for them to guess who you are based on your usage patterns alone, e.g. what you look at, what you type, how fast you type, and so on.

So yes it’s possible to get an anonymous account on Facebook, but it’s inconvenient, and probably runs contrary to your goals on that platform. Which is why I recommend to not use such platforms at all, and instead keep in touch with your friends via secure and private open source based messengers like Signal or Threema (Libre?) or any Matrix client. After all, services like Facebook are mainly for Meta to sell data about their users, that’s their primary business model (they almost don’t sell any products or services, they almost solely sell data about their users, that is basically the only way they make (a lot of) money. Well, and ads of course, but that’s closely linked together because the ad industry wants personally targetable ads, so it needs user data to better place targeted ads). There also isn’t a real need for such platforms anymore, because you can connect to friends in other ways on various other platforms, the only hard part is getting your friends to stop being lazy and switch to something more secure or private. When Facebook got big, it might have been the only social network of its kind, but these days tons of other, better options exist. So get your friends and contacts to move to something else. Or if they don’t want to move with you, then maybe they aren’t that important to you to keep in contact with.

Using Facebook in any other way, shape or form (e.g., using their app, or using their website with your real IP address) is very likely not going to be anonymous to Meta. They’ll automatically receive your current IP address (which on its own might already be enough for them to establish a link to your person, since they also have trackers in place in lots of other apps and websites), and on top of that various information about your device or browser (which, again, can be a key factor to link your current usage data to your person), if you use their app (which you shouldn’t ever do) they get even more data on you (not just you, also your contact list, nearby devices, and things like that), because those apps require so many permissions and have so much tracking integrated that it’s a whole treasure trove of information that’s being sent about you and your device, and they’ll interconnect all that data with the other data they’ve gathered about you or your device(s) in the past (which, as a rule of thumb, will always be much more than you think they’d have). An app with integrated tracking is always more harmful to your privacy than using their service from within a web browser, because the app can read much more data about your device compared to the web browser. (But be mindful that some web browsers (especially the proprietary ones like Chrome, Edge and Opera) also have quite a lot of tracking capabilities inside them.) So using Facebook in a somewhat normal or convenient way and at the same time wanting to remain anonymous to them is basically impossible.

Also, you’ll never be anonymous to government-based mass surveillance (who are collecting almost all network traffic, constantly) when you use your real IP address online. Anywhere. Your real IP is always connectable to your real person for them (also in retrospect). Even if they can’t look into encrypted communication data, like the contents of chat messages or what you did on a specific website, they can see the metadata, among that is which hosts you contacted, and when, as well as more unencrypted details, and such metadata can already be very revealing. To quote the ex NSA chief “we kill based on metadata”. Protecting yourself against commercial-based surveillance by companies like Meta is more realistic to achive (at least partially), because it’s easier to avoid or evade commercial tracking (by blocking all or most of their tracking methods like app-integrated trackers, tracking Javascripts and cookies on countless of websites, and so on) than it is to evade someone who’s sitting directly at all relevant network cables AND buys additional data from companies. Lots of easy-to-use tools exist to counter or limit commercial surveillance, like ad/content blockers, blocking host lists, PiHole, ad-blocking DNS servers, open source software and operating systems (because they are almost always free of trackers and surveillance tech), and things like that. It all minimizes your exposure to these data hoarding companies. And the less data you transmit overall to such companies, the better. But if you also want to protect yourself against any government-based mass surveillance, you’ll have a much harder time than that. You’d need to always use different IP addresses (again, via Tor or VPN etc.) and avoid having anything leak out that can connect your other IP to your real IP. Which is hard.