When using resurrection spells like Revivify or Raise Dead, it's obvious that you return to life as you were before you died - as in, same body, same age, etc.

However, Reincarnate, the druid's resurrection spell, fashions a new body for the target. Since spells only do what they say, we have to assume that you will be as old as you were before you died - but are we talking about physical age or mental / actual age?

For example, what happens if you're a 90-year-old human who gets reincarnated as a member of a longer-lived race, such as elves?
Are you now an elf with a physical age of 90 years (and therefore a pretty young elf)? Or are you the elven equivalent of a 90-year-old human, and therefore also suffer from the disadvantages of old age (whatever they may be for elves)?

Yes, a reincarnated creature has the lifespan of its new body

5e official rules designer Jeremy Crawford answers this question here:

More about reincarnate: if you're a 200-year-old elf, you're still 200 after being reincarnated. But really the question is about the body. You get a new adult body—not young or old—appropriate for the body's race, so your inner and outer ages can be mismatched.

Which age determines when you die of old age?

If your soul and your body have different ages, bodily death is tied to your body's age, not your soul's.

So that means reincarnate can be used repeatedly every time a creature dies of old to let them live indefinitely?

That's correct.

This fits with the spell description, which has no exception preventing it from working on creatures that die of old age, and which states that it creates a "new adult body" for it. Your soul and memory are the same, but it's a totally new body - and if the new race has a longer lifespan, the creature now has that lifespan. (Though how often do adventurers actually live long enough to die of old age?)

Yes. The spell changes the creature's racial traits.

From the spell's text.

The reincarnated creature recalls its former life and experiences. It retains the capabilities it had in its original form, except it exchanges its original race for the new one and changes its Racial Traits accordingly.

The creature is now an elf. See the PHB description of elf to assess the effects of the change.

A Timeless Perspective

Elves can live well over 700 years, giving them a broad perspective on events that might trouble the shorter-lived races more deeply.

This provides a rich role play opportunity as the character gets used to being an elf.

Elf Traits (PHB)

Age. Although elves reach physical maturity at about the same age as humans, the elven understanding of Adulthood goes beyond physical growth to encompass worldly experience. An elf typically claims adulthood⁠ and an adult name around the age of 100 and can live to be 750 years old.

RAW support for this position: the elf trait specifies the longer age before death.

Dev tweet support here, if needed.

{Crawford} If your soul and your body have different ages, bodily death is tied
to your body's age, not your soul's.

So that means reincarnate can be used repeatedly every time a creature
dies of old to let them live indefinitely?

{Crawford} That's correct.