An instructor must keep a copy of the documents for five years that are
used to prove citizenship or make an endorsement in both the
instructor's logbook, or other record used by the instructor to record
flight student endorsements, and the student's logbook with the
following:....(endorsement example)

The sentences are structured poorly (what else do you expect from
lawyers) but I read it thusly (emphasis mine):

An instructor must keep:
a copy of the documents (for five years) that are used to prove citizenship

=OR=

make an endorsement in...

both the instructor's logbook,
(or other record used by the instructor to record flight student
endorsements,)

and the student's logbook

WITH the following:....(endorsement example)

That said... were I the CFI, I would certainly photograph my student's
passports, driver licenses, and any other information I thought the TSA
might fault me for not having. If this generates a political backlash,
so much the better.

Jose
--
The monkey turns the crank and thinks he's making the music.
for Email, make the obvious change in the address.