Well how to "ensure that all functions and model methods called by the request use this connection" is easy. You pass the connection into your api as with any well-designed code that isn't relying on global variables for such things. So you have a database session object loaded per-request, and you pass it down. It's very easy for model objects to turtle that session object further without explicitly passing it because each managed object knows what session owns it, and you can query it from there.
db = request.db
user = db.query(User).get(1)
user.add_group('foo')
class User(Base):
def add_group(self, name):
db = sqlalchemy.orm.object_session(self)
group = Group(name=name)
db.add(group)
I'm not recommending you use that exact pattern but it serves as an example of how to grab the session from a managed object, avoiding having to pass the session everywhere explicitly.
On to your original question, how to handle multi-tenancy... In your data model! Designing a system where you are splitting things up at that low of a level is a big maintenance burden and it does not scale well. For example it becomes very difficult to use any type of connection pooling when you have an arbitrary number of independent connections. To get around that people commonly use the SQL SCHEMA feature supported by some databases. That allows you to use the same connection but have access to a different table structure per session. That's better, but again managing all of those schemas independently should raise some red flags, violating DRY with all of that duplication in your data model. Any duplication at that level quickly becomes a burden that you need to be ready for.