Way to add a background color in Apple notes

This isn't exactly what you were asking for, but perhaps it is close enough.

Just below the menu place where you bring up the color picker (Format->Font->Show Colors) are two other menu items: Copy Style and Paste Style. This is what worked for me:

1. Select some text in your note
2. Manually set the color, like you do now (via Show Colors)
3. Set the cursor anywhere inside the colored text (don't select any text, just put the cursor there)
4. Choose Format->Font->Copy Style
5. From now on (including across relaunches of the Notes app) you can select the text you want to color and simply use the keyboard shortcut for Paste Style!

Hope this works well enough for you.

Not sure when this started, but I found something relevant to this question just today. I really hated the color of the links in Notes, too, and had come across this question months ago and wanted to return to add my discovery:

Link color is mostly connected to highlight Accent color in Sys Prefs. I'm running 10.14.6 and using Dark Mode. I recently changed my highlight color from grey to orange (the orange looks good against the slate color of dark mode). I just happened to open a note containing a long list of links. It took a second to realize, "I can read these links!" Then, "I can read these links because they're orange!" I tried changing the highlight color to blue and the links turned yellow (against Dark Mode that now works) not blue. So mode also plays a part in determining link color in Notes.

EDIT: The change in Accent color changes the link color. Changing the Accent color might also change the Highlight color but they don't necessarily have to be the same.

System Preferences > General > Accent Color

Hope that helps someone.