This was not an issue with LH and other partners until Lufthansa began charging for seat reservations for tickets issued beginning April 28 of this year.
Not everything in that FT thread is relevant to your situation, but it does include some who were similarly surprised after buying tickets.
Lufthansa began charging for advance seat reservations for lower discounted economy fares earlier this year.
And apparently, it can only charge for the seat reservations if the ticket is issued by Lufthansa (ticket number beginning with 220).
Is that what I could use to do an online check in with them?
I'll admit I became even more confused after reading through that long Flyer Talk thread.
I'm now confused because United told me that they don't have access to Lufthansa's seating software, so I am guessing that I might not even be able to do that online check-in on the return trip.
When I called Lufthansa, they did give me a Lufthansa confirmation number.
The wiki at the top of the ft thread says that UA can't process payment for the advance-seat-res fee.
Apparently, you can't pay that fee directly to LH because UA issued the ticket.