Squishy polytheism is gaining ground, but a lot of people seem to think that the line between hard and soft polytheism is a hard line - either all gods are the God and all goddesses are the Goddess, or every single god is totally different and unrelated. I am a firm believer in middle ground, however.
I was on the phone the other day and I thought of another metaphor for how it works: when you talk to, say, your insurance agent, you are talking to a person. She is an individual, here to help you with your car insurance. You may work with a different insurance agent for your home insurance, or you may give both areas over to one agent. That agent is empowered to deal with you by the company she works for, and she is the face of Some National Insurance Company. That doesn't mean that she is just a mouthpiece for that that company, and it doesn't mean she's interchangeable with any other agent. You have a relationship with her; she's more likely to let you know about something that'll help you out, and you're more likely to give her a call when you need more insurance.
Both the Insurance Company and the Agent exist.
To put it in somewhat nerdier terms, Phil Coulson is an agent of SHIELD. He is not SHIELD, and he doesn't have access to all of SHIELD's tools and options. But he has access to quite a few, and it's a lot easier to convince one agent that you need the Avengers than it is to file form #2429B in triplicate with SHIELD and see if they get back to you. Clint and Natasha and Maria Hill are all SHIELD agents. If I needed something shot with an arrow, it's going to be more efficient to talk to Clint and ask him to shoot it with an arrow. But if Clint doesn't know me from Adam, and Phil's on my Christmas card list, I might call Phil up and ask him to ask Clint to shoot something with an arrow for me. (Or hell, Phil might decide that shooting with a bullet is good enough and do that for me instead.)
And even though Clint and Green Arrow are both really good at shooting things with arrows, I think Clint would be kind of pissed if I called him Green Arrow while I was asking him to shoot things for me.
I think I'm wandering a bit, but hopefully the idea got across...