Soulmarks aren't always a good thing. People say they stand for a grand destiny, but Tony is a fan of evidence over folklore. And evidence states that your 'grand destiny' could simply be your ending. Death or something worse.
Tony thinks his soulmark just represents his death. It certainly doesn't mean that Bucky and Steve are his soulmates.
James Buchanan Barnes: formerly the Winter Soldier, formerly Captain America’s right-hand man, formerly a sergeant, formerly a dock worker, formerly Stevie’s best friend.... currently a glorified prisoner of Prince T’Challa.... had trouble wrapping his head around all those former selves. He spent too much time thinking about all the bits of him that he kept gluing back together to pretend he was a person anyone would want to keep. He spent even more time picking at the cracks, pointing the flaws out to himself. Only thing he did more than that was make sure no one else noticed how far he was from human.
So, James Buchanan Barnes, who didn’t know what to call himself, who didn’t know how to go forwards, agreed, and moved back into the tower where his best friend lived with a husband orphaned by the Winter Soldier.