He's trying to form a closer bond with her
As per Tyrion's own words (not exact as it is from memory):
I suppose the longing looks your way are in hope of forming a strong military ally.
It also appears that Dany has an attraction towards Jon as she waits for him on the wall, looks worried at the state he's in and sits by his bedside with him among other things.
With all this they both seem to have a romantic attraction towards each other. In this case to further the bond would you try to call her, her full name:
Queen of the Andals, the Rhoynar and the First Men (claimant), Protector of the Realm (claimant), Queen of Meereen, Khaleesi of the Great Grass Sea, Mother of Dragons, The Unburnt, Breaker of Chains
Or the name close friends do:
Or would you try and make your bond seem closer by using a more personal touch:
I'll leave that as an open ended question but I know which one I'd go for.
OOU it's a set up for the next line as well:
How about My Queen.
Regarding the edited question:
If I told you my name was Joseph, what nickname would you pick? Joe, Seph, Jose? I think you'd go for Joe. The same applies here, he just shortens her name to Dany.
He also doesn't know someone else used to call her Dany; it's a nickname he has chosen for her though I presume he'll choose another one as per her reaction.
Lastly, he doesn't know that she won't burn him, but he's counting on her not being bat-shit crazy and her liking the name. To form a closer bond with someone, risks have to be taken, and I'm pretty confident that him calling her Dany is one of the lesser risks in such a situation.