How to create borders wrapping text

You can make and save custom stroke styles in InDesign. Here is a link to a tutorial:

https://www.rockymountaintraining.com/adobe-indesign-creating-new-stroke-styles/

In addition, you can add type on a path and select a symbol font, like Zaph Dingbats, although you would need a separate text box for your copy.

"So is there a way to set an Illustrator file to be used in a paragraph style, so it adapts the space of the text?"

No. As Derek points out, you can set a text frame (and any native stroke, rules, or shading assigned to it) to auto-size as you add or remove text, but a Placed graphic from Illustrator cannot be made responsive in the same way.

If you want a text box that ajusts to the amount of text, have a look at: Object > Text Frame Options > Auto-size.