I don't yet have a definitive answer, but I have contacted the CBP and it appears they don't have an official policy on this. The requirements seem to be at the discretion of the specific enrollment center.
Here is their first response:
You can log into your account and register any vehicle you wish.
If you register your vehicle in SENTRI as part of your initial
application, there is no fee for registering your vehicle. If you wish
to register another vehicle, you must pay a $42 vehicle registration
fee. There is no longer a requirement to visit an Enrollment Center
for a vehicle inspection when registering a new vehicle.
You can register up to 4 vehicles for use in a SENTRI lane, and you
can have up 8 SENTRI members registered to a vehicle. Approved SENTRI
members are able to use the Global Entry kiosks when re-entering the
U.S. by air and TSA Precheck, as long as they meet the citizenship
requirements for Global Entry and their fingerprints and passports are
on file. They are not, however, entitled to any benefits under the
NEXUS program through their SENTRI membership.
This seemed to be a bit of a generic answer to me, and although they mention "any vehicle you wish" in the first line, they then go on to talk about "your vehicle" rather than any vehicle, so I asked them to clarify.
Their response to my request for clarification:
You will need to call the enrollment center and ask the officers for
Sometimes individuals have said they were told
they could register vehicles with only a letter from an owner and
others have stated they were told the owner had to be present when
they registered the vehicle.
I will include an update once I hear back from an enrollment center or two.