DL sells some Asian destinations using SQ/ANA for connections. The SQ/ANA flights dont earn any DL miles.
You can have both on file. Since you earn nothing for DL flighrs in SQ/ANA (or vv) program there's no issue if system sends to wrong program.
Issue comes up when you can earn in the other program. In that case system sees you already got credit and denies. (This happens frequently with DL SM and AF/KL/KQs FB program)
Do note some programs care about ticket stock. (DL doesn't) Thus make sure itll earn if on DL ticket stock.