Both the ability to edit the background color of a tooltip and the ability to resize a tooltip are not functionalities currently built into the product.
To voice your support for the inclusion of formatting the background of a tooltip in a future product release, add your vote to the following Community Idea: https://community.tableau.com/ideas/5357
Another option to provide suggestions for product features is to submit an idea through our community: Space: Ideas |Tableau Community
One workaround to both functionalities may be to conditionally format/color text in tooltips as described here: Format Tooltips for Conditional Dimensions | Tableau Software
Hope this helps!
Well, you can hack around with the tooltip a little bit.
In the attached (V 8.2, so nothing special that 9.x or 10.x can do is needed to do this) I have made there calcs. One sets a value if the mark is the smallest. One sets a value if the mark is the biggest. One sets a value if the mark is not the smallest or the biggest.
Then in the tooltip I put the three calcs. Only one will display, based on the size of the mark. And I messed with font size for the three calcs. And (here's a hack) I added more characters and colored them white (so they won't show) to increase the size of the tooltip box. (Highlight the entire text of the tooltip to see what I did in there).
It's just a hack, but it shows a way to change the size of the tooltip box. Maybe it's something you can use for your purposes.
superstore -- tooltip size.twbx 976.2 KB