I just financed a model S today so I can give a little insight. Tesla doesn't have their own bank like Toyota any other does. When ever you apply through their website, they shoot your app to their partner banks they are working with to get the best bid. Alliant Credit, US bank and I can't remember what the other two were. So there's nothing special or different tesla does far as Financing because they use a regular credit union. Then you wait till the tesla rep calls you to to tell you what you got approved for based on only those few banks.
If you don't get approved through any of them. No worries. Just like any other car, you can use what ever bank you want of your choice and try your luck there. You don't even have to apply at all using Tesla's site. You can go straight to one of their partner banks and apply exclusively with them.
Now, this is how I did it.
I applied to PenFed credit union because 1. We used them in the past and got a great rate 2. They use Equifax Fico 9 scoring which was our highest scoring (810) got approved for up to 60k with 110% LTV. But we needed more since our model S came to 90k. So we applied directly to one of Tesla's partner banks Alliant credit union. They used Transuion (not sure what scoring model) and they rated our credit at 730. But approved us for up to 85k with 90% LTV.... lower score but more money then PenFed? I don't get it either lol
So we went with them. We only used 80k of their loan and paid the rest from our account. Side note: they don't accept credit card payments, only check or bank transfer for the remaining.
Again, you can use any bank or credit union you wise. I do believe there are a few banks that do Tesla loans for some reason so make sure you call the bank and ask before applying. The more cash down, there better because some banks don't like to finance 100% of it.
Also, PenFed approvals are good for 90 days! Alliant is good for 45 days before you have to reapply. Hope this helps!