Is it even possible?
I haven't tried it yet, but some people say it is not possible to send mail to characters of your own. ESO is forcing you that way to use your bank and spend gold for bank slots.
What is true?
That's correct. You share the same bank amongst your characters.
You can also start a guild and add 10 people (I believe it's 10) to have access to a 500 slot guild bank. You won't be able to put Bound items into this bank though.
XboxOne - NA - Gamertag: IlI SYNN IlI
CP 690+ - DC