Rephrasing the above:
Text styles are great!
Once text has a paragraph or character style, changing the style also updates all text with that style.
Text styles are part of the document; I don't have a bunch of different "Heading 1" styles from different documents competing for space in the menu
Text styles can be based on other text styles, and don't have to define every attribute: a paragraph style could be "Whatever Body Text is like, but green."
Object styles... are not like that.
There does not seem to be a way to update an object style once it's been added to the panel (Screenshot 1)
It's more like a "Paste Format" palette.
Object styles seem to be app-wide, not document-scoped
There's a "Default" grouping menu, but I don't see a way to add other groups (Screenshot 2)
Object styles cannot inherit from each other
Suggested style-eligible attributes
Everything that can be set in the Color/Stoke/Constraints/Quick FX/Transform (etc.?) panels
Probably also Text Frame settings
(I would not have thought of this, but Paolo's "Picture Frame Styles" post particularly called out position-on-page as a useful attribute)
Screenshots:
Can't update object styles
Can't add new groups?
Other Threads
This has also been suggested as "Picture Frame Styles":