Changing color of container in a repeater

Can’t find any → STYLE ← attribute on → CONTAINER <—

But as you already mentioned →

I see there's background attribute for the container, but looks like I can only change src (i.e. image is my guess). 

Yes you can use it if you are tricky.
And if you are very tricky —> you put another element, which supports the STYLE-ATTRIBUTE, you put it into your CONTAINER setting it as TRANSPARENT (OPACITY = 0%).
And control it by CODE.