No direct connection exists. DS and Wii are separate platforms with different hardware, OS, and no built-in data link. You cannot "sync" them like a phone and watch.
No "DS to Wii cable," no Bluetooth pairing, no shared account sync (Nintendo Network ID / Nintendo Account came later). Each console stands alone.
If you want cross-device play, Switch + mobile (some games) or PS4/PS5 + Remote Play are modern equivalents.