The doc page says that the user object returns an age property (for those users that specified an age).

But the API  no  longer  does.   ^{_{(Plus spot checks of users that had age in their profiles at one time.)}}

Note that:

1. Age was removed from users' (site) profile pages in April 2015.
2. Network-wide profile pages dropped the Age display in April 2017.
3. Sometime in the last few years, the site stopped allowing you to enter your age.
4. The Data Explorer (SEDE) stopped reporting Age yesterday (June 2018).

Update: Nick Craver says that the age property cannot safely be removed, but that it will now always be null (which means that it will not be present in any API results).
So, from the user's perspective it is removed, but it will still be present as an obsolete option for filters.
The documentation still needs to be corrected.