It's not currently possible to edit the background color of a text tile within the text tile itself. I see how it would be a useful functionality though, and I'll be sure to share it with the product team.
In the meantime, I have an idea that I'd like to share. Since you can use html to format results, I imagine you could define a dimension for a string literal and use html to format it. If the string literal has a lot of characters, I'm not sure how line breaks would be handled, but there may be ways to accommodate that depending on the dialect.