Two strategies: What the really rich do, and get real fake identities
(1) What real wealth multi-generational families do?
The short answer is to google "Single Family Office".
Wealthy multi-generational families have a organisation dedicated to serving their personal and corporate needs - from PAs, security and travel, to property management, special projects, PR, charitable giving, investments etc. Or whatever they wnat.
They already plan with a very-long term perspective, just for real multiple generations. With offices in multiple countries, resouces squirreled away or perpetually "in transit" etc. This bit of the problem has a real-world solution in place.
The only challenge remains creating new legal identities every generation.
Seems to me that with planning it wouldn't be hard to double-up some birth registrations.
Take the mother-to-be to two doctors (perhaps in different countries). After a home birth register the birth twice. After that it is mostly paperwork. Keep the parallel papertrail going. There may be a few occassions when a body must be presented but likley can be managed in the same way. Criss-crossing jurisdictions would make it much easier too.
When a suitable age, the identity is adopted by the immortal.
Repeat every 40 years or so.
(2) Certain agencies routinely create false "legends" for people
Intelligence agencies, witness protection schemes, etc routinely create robust fake identifies for people using REAL government documents, accounts etc. So one needs a good relationship with some of these people
Would have to think through whether certain select people would KNOW about you and be aware of what you were doing, or whether the Chinese Walls in such organisations would mean they wouldn't realise there was a sequence of identifies with ever-later birth dates.
If you are actually invulnerable-immortal you could make oneself VERY useful to these people (the film "The Old Guard" is a lovely exposition of this). If you are merely unaging-immortal there are probably still ways of building such a relationship (wealth, knowledge and patience!).
The biometrics problem
An emerging problem would be the ever-increasing use of biometrics to establish identify leading to double-matches.
So far things like passport biometrics are primarily confirmatory - you present an ID with pre-established biometrics and it is confirmed that you are a match.
With some - police fingerprint databases say, as opposed to a simple fingerprint lock - the systme is used prospectively to identify the person without a reference identity to confirm.
I am not sure how certain these matches are, vs what size database. But over-time the chance of them popping up in a list of possible matches with two or more of your identieson them would seem to increase.
Of course we don't know how immortality works - finger prints wear out a bit over time. Perhaps immortal fingers will change enough over decades for this to be a problem. Perhaps their DNA is mutable as a consequence/cause of their immortality so it won't match so well (but are their genetic markers for immortality that someone could notice). etc