Does switcheroo target? So Twilight Shroud prevents it?

Yes — Switcheroo targets, so Twilight Shroud stops it.

Switcheroo chooses two units at the same battlefield as targets when you finalize it (Rule 355.5–355.8; FAQ #8679, #11522), not on resolution. Twilight Shroud makes a friendly unit "can't be chosen by enemy spells and abilities this turn," which is untargetability (Rule 757) — so that unit simply cannot be one of the two units an enemy Switcheroo picks. If Twilight Shroud is played in response to a Switcheroo already targeting the unit, the unit becomes untargetable and the spell mistargets on resolution (Rule 758.1); since a swap requires both units (Rule 433.1), Switcheroo then does nothing to either unit. Same precedent as Ruin Runner vs. Switcheroo (FAQ #9313).