They really don't, at least not in a way where their genomes would mix in equal parts, or be concatenated.

The closest to merging is probably mitochondria, which probably were single-cell creatures ingested by some very early (single-celled) ancestor of us and, instead of being devoured, ended up in a symbiotic relationship within its host. The mitochondria to this day have their own genome and function a lot like a cell-within-a-cell. Same for plants and photosynthesis.