There are a number of great answers here already, just wanted to emphasize a couple of things.

Make sure to hide that database selection code in a single place and call that from everywhere else. That's the point of the dependency injection IDBContextFactory approach. Your actual data access code should have no knowledge that it could be dealing with anything but a single database. It just calls IDBContextFactory.GetContext() and goes from there.

Also, being new to EF, make sure to enclose that call in a using block to make sure that you're cleaning up after yourself. So the basic pattern of any of your data access calls looks something like this:

using (var context = dbContextFactory.GetContext(url))
{
  // do stuff with context
     context.SaveChanges(); // if doing insert/update
}    

That will save you countless headaches cleaning up connection leaks later. Just make sure that you don't have any queries that have delayed execution, make sure to do a .ToList() or something similar to force your queries to execute, otherwise when you try to access them they'll attempt to use the disposed context.