Co-op mode requires at least two active players to enable, but one thing to note is that this achievement only requires you to play 10 songs in co-op mode, it doesn't require you to score well.
What I did for this was started a song, and at the dancer selection screen, had my wife stand next to me, enabled co-op mode then started the song. At that point I let the song play out while her & I did chores around the house. I'd end up finishing with 0 stars, but still counted. Rinse & repeat 10 times and the achievement is yours.
It's also worth noting (it's in the achievement description, but for the record) this is for doing 10 different songs. So sorry, you can't just do You Got a Friend in Me (as the shortest song in the base game) over and over.
Also: it doesn't matter if a song is solo, duet, trio, or quartet, all can have co-op enabled & count (for solo it just means that both dancers would dance the same part).