The way one would normally do this in InDesign is to put any background graphics onto a Master page and set your margins there to the "inset" you want. It's just as much a time-saver, if I understand your situation correctly.
That's how I would normally do it too, but I need to shorten up the text frames on certain pages.
It is confusing how I'm describing it.
I guess the simple answer is a NO. You can't place a photo in the background with text, all in the same frame like you do on a website DIV.
You can group objects, then cut and paste the group into a single frame, if that helps.
It's a bit tricky but you could anchor the background image behind the text. So:
An illustrator file and some text:
Cut the background, place your cursor at the beginning of the text, and Paste:
Open Anchored Objects Options and choose Above Line with enough negative Space after to position the background:
The downside is it gets difficult to edit the text because the anchored image gets in the way, so you would have to use Story Editor to easily make text edits: