I don't know about spawns being marked in the map editor as belonging to any particular team.
The lobby has 3 ways of assuming which team a spawn point should be one (top vs. bottom, left vs right, or odd vs even). Those are the options to choose between. All 3 of those options assume there will be exactly 2 teams. So none of those three options work for you. That's why it's necessary to set teams manually for a 3-team match.
It is considered good practice to support odd-vs-even team assignments if someone is making a map that is balanced for competitive multiplayer games with 2 teams.
But for a 3-way map, you wouldn't have that. I would just group them up like: spawns 1-3 are for team 1, spawns 4-6 are for team 2, and spawns 7-9 are for team 3. That way, in the lobby, it's easy to see the teams because the players on each team are grouped together (the first 3 spawns will be team 1, etc.)