Firstly a modern science take on this.
Your question suggests an infinite time universe. Which inevitably implies a recurrence of the physical structure that is "you" right now.
Just change the assumption to infinite space. What happens then is that your concern about recurrence "billions of years later" becomes RIGHT NOW.
Just as for pi:
the digit 1 repeats at the third place, 6 nines occurs in the 762nd place and so on... every conceivable (finite) pattern presumably occurs somewhere or other.
Other famous analogy: Enough monkeys will produce all literature
So likewise you ALREADY "multi-exist" in this infinite universe. Maybe even you already infinitely multi-exist!! Does it change anything in your experiencing?
I think not! SO why a billion year worry ?
However this is not a satisfactory answer (to me at least)
Your question is an important piece, maybe even the foundation-stone of all the dharma religions: Buddhism, Jainism, Sikhism and the original root Hinduism.
You may naturally find the style and language odd due to cultural factors.
If you cross that barrier you would find an anormous amount of thought given to this by a wide spectrum of saints, sages, etc across a long stretch of time to this.
The language barrier is there, they call it karma कर्म. (Actually the dharma religions should better be called the dharma-karma religions. Where roughly dharma means the law we 'do' and karma means the law done to us.) No God is handing out reward or punishment. We ask we get. The guiding directing force is our wish.
Its famously known as tanha in buddhism or trishna तृष्णा in sanskrit
This word karma is usually kept as is because it has a meaning so deeply rooted in the dharma traditions that it is hard to take out! Still let me try!
Firstly karma means action/work/doing. The most ordinary meaning
Secondly somewhat more widely karma means action AND its consequences
Thirdly in its most generalized setting it just means causality. Alternately law as understood by modern science
Take the law of conservation of momentum.
The interstellar comet borisov split into two as it passed our solar system .
The momentum of the two pieces will remain the same as the momentum of the original but its identity is a question mark: Which shall we call borisov?
Reincarnation or karma is similar (at least in the dharma religions)
Everything in life goes by this.
Because we wish to eat we cook.
Because we wish to keep eating and so on we work to earn.
Many people do terrible things because they are consumed by their wish (tanha). Rape, Robbery, Murder etc. All are driven by uncontrolled wishes.
And so what happens if we die before our wishes are fulfilled?
The identity disintegrates but the momentum continues.
So like Borisov when it splits we may not be able to properly call it borisov because now there is borisov-a and borisov-b. And like the momentum that remains the wishes that we sent out will fructify even if we dont remain to receive those fruits! This inexorable terror is the root problem that all Hindu mythology or Buddhist psychology etc are trying to address.
So in summary the message you will understand (from the dharma religions at least) is stark and simple: If your wishes dont end before your life, reincarnation (Nietzsche called it recurrence I think) will continue eternally meaninglessly.
 Dharma by Veena Howard https://www.bloomsbury.com/us/dharma-9781786732125/
 Six nines in pi https://en.wikipedia.org/wiki/Six_nines_in_pi
 Galactic comet borisov splits into two https://www.universetoday.com/145603/interstellar-comet-2i-borisov-appears-to-have-broken-in-half/
 tanha https://en.wikipedia.org/wiki/Ta%E1%B9%87h%C4%81