I don't believe this is possible directly (It's possible to have an anchored object assigned a paragraph style as part of an object style, though)< but it might be doable with a script, though that is not automatic, and I'm not sure it wouldn't duplicate anchored objects each time the script is run. You can ask over in InDesign Scripting and try to explain in as much detail as you can what you want to do.
As Peter says, a script might do it but a simple way would be to create on object the way you want it and create an object style from it.
It would be pretty simple to then create them on the fly.
@gabbi – no script necessary. You can do it with a GREP search and replace action.
Step 1: prepare your anchored object in a separate text frame
Step 2: copy the only character in that text frame
Step 3: Do a GREP search/replace:
Search for the appropriate insertion point where you like to add the anchored object.
Replace that with the contents of the clipboard.
//EDIT: added "frame" to "text" in Step2.
Message was edited by: Laubender
Here an example for a GREP search/replace:
Search for the beginning of every paragraph + the first character:
Replace with the formatted contents of the clipboard + the found character: