What does the red background on an item mean?

Based on my observations, your theory appears to be correct. Spinning a disc that is on a gym that your team controls will give you an extra item. That item will be indicated by the colored background that matches your team's color.

For reference, I am on Team Mystic. When spinning the disc at Valor and Instinct gyms, I did not receive a "special" item, but I have at the Mystic-controlled gyms I've spun.

This is also being mentioned in other places, such as this thread:

To confirm my suspicions, the team bonus items are highlighted with team color, correct?

It seems so, yes. My team bonus was highlighted red, and I'm Valor.

Flavor thing: as a Mystic player, my team bonus text and background on the bonus item are blue. Can Instinct and Valor confirm their text/background colors on bonus items?

yes instinct here, yellow.

Valor, the bonus item text is red.