No they don’t. It’s a nice sentiment, but not Buddhist at all.
One of the central tenets of Buddhism is Anatta- or the concept that none of us have a “soul” in the sense of a permanent, unchanging self.
If that were true it would mean we are all eternally, irrevocably separate from each other- essentially solitary.
Buddhism rejects this falsehood. All separation between you and others is simply a delusion of the mind, unable to perceive true reality and non-duality.
You as as equally “soul mates” with every single human and living creature as every other. And they are as equally you as you are them.
A lot less romantic, but a lot more True.