How to fill gap between garage floor and driveway
A very common gap size for concrete swimming pool decks, check a local pool store for wood, plastic and rubber expansion joint fillers. Thinner Concrete Pavers could also be cut into strips and dropped or stuffed in.

Otherwise, there's no problem with filling (and ramping, if desired) the gap with concrete or cement. The filler won't have any interaction with either slab, since they're fully separated and new cement bonds extremely weakly to old cement.