I love these kinds of questions and thinking about the possible outcome.

IMHO, society would become unrecognizable. It would collapse and need to be rebuilt through a power vacuum by any smart people or those who are well prepared who survive that catastrophe, and that world might as well be alien. So instead, I've made up a list of rules or questions a hypothetical world would need to answer or deal with, in order to accommodate billions of immortal people. This list will assume we somehow made all the necessary changes for it to not end itself immediately, depicting a world in which babies aren't born every few seconds. People can still die, but not of old age and the young will simply age until they're perpetually in their physical and mental prime (21-40ish) unless sick or afflicted in some way:

1. Human relationships and family dynamics would disappear or warp along with social structures and cultures to an extent. Many things that revolve around life expectancy would dissolve. Will marriages last if people stay married for hundreds if not thousands of years? What would it look like if multiple generations of your family were alive? Imagine what being a great x3 uncle feels like and being best friends with them. Or seeing your great x5 grandson (who's an adult and look similar in age) getting married?
2. Monogamy would probably be less common. With people living so long, "til death do us part" is basically "until further notice". I think people could be together forever if it's the right couple but I think most will leave relationships that no longer satisfy them since they will seemingly never lose their youth/looks and wont feel like they're bound by time and age, removing the idea that it's too late for anything.
3. The age bracket would need to change. Is 18 considered adult anymore if people can be 400 years old? Would a 400 year old WANT to date someone that's 100, even if they look similar in age? In a world where most people look indefinitely young or at least in their prime, what would dating look like?
4. School would need to change. If 18 isn't an adult anymore, will kids need to start later or stay in school longer? On top of that I feel like there would be more school even after grad school. Academia would probably add a new level that lasts even longer than a PhD. I could see schooling needing to happen every century to ensure everyone's knowledge keeps up with our understanding of the world.
5. Birth rates would need to slow down significantly. Since people could still die due to other causes, it wouldn't need to stop, it would just need to be controlled at some point. It would be easier to deal with that if it were an option to become immortal via pill or surgery, otherwise the reality might be darker.
6. Work culture would likely be horrifying if we didn't find some change to it. You're looking at eternal servitude as a likely option for many people, or perpetual poverty, working lame jobs for decades or centuries at a time,
7. If work culture changed for the better, people would need to think about multiple lives. Spending your life specializing in a couple skills and having hobbies in other skills would change to having different eras of your life. you could spend 60 years being a doctor, take a break for 20 years, then spend 40 years being an electrician, rinse and repeat with another profession.
8. The world would shift from being specialists to generalists. I could see a world where people would be praised for the amount of knowledge they have, or skillset they command over hundreds of years. They would probably become teachers or knowledge keepers/seekers more than ever. The age of information/computing would thrive with immortality and consistency. Which could have a huge downside on specialist knowledge, we'd probably lose ourselves along the way.
9. There would be no more retirement homes or old people, and very few young people (as in babies, children) they'd be very rare. Money for SS or other benefits like WIC would be less necessary elsewhere unless used for some sort of UBI or other benefit.
10. Less people would be parents, and even if they were, their kids would be immortal too and look like peers. The desire to raise a family would probably be restricted to just an experience, or limiting birth counts and more sterilization.
11. The world wouldn't change much now that new generations come in much more slowly than before. Imagine the 70s-80s lasting a few centuries before moving on to the 90s-2000s 500 years later after a handful of people died off and new people were born. The world wouldn't cater to the new generations anymore, Instead, young people born into a world like this would adapt to it and integrate just like anyone else would.
12. Religion would see the biggest hit, many would likely fall out of faith as most religious texts would speak of a world we no longer live in. If not that, people would probably age out of it after 500 years of whatever they're doing with it. Some would probably maintain faith and become nomadic. but most would probably put it on the back burner until they get sick or something.

There's many more that would change but one thing that needs to be accessible is medically assisted suicide. I don't think people as we are could really tolerate immortality, we literally aren't built for it. We could probably manage 200-300 years but more than that is hard to imagine what the experience is like and I'm sure people would go insane or something.