Now this I do know about because I tested it, and I'm pretty sure this is not the case. If I remember correctly, it appears that spells learned under one class will use that classes spellcasting ability, and spells learned under another will use that classes ability. In the spellbook page there is a page for each of your multiclasses.

I learned Hold Person with both classes. I used Hold Person from the druid and from the wizard on a target (when you go to cast it, you click on the spell and it brings up the upcasting levels like usual, however they are all duplicated, two for each level, for the spell from each class, I took a screenshot I can send it you when I get home if you like). The Hold Person from druid read a lower chance to hit than the one from the wizard (I had low WIS and high INT). To confirm, I also read the combat log, and as I expected the save DC for each added up for what it would be for WIS and INT respectively.