For a Kotlin/Wasm target, forwarding logs back to the IDE doesn't seem to be possible yet. Kotlin/Wasm is still in Alpha, and some features are limited. I'll quote the Official Kotlin Documentation: "Currently, debugging is only available in your browser. In the future, you will be able to debug your code in IntelliJ IDEA."
I'm less familiar with Kotlin/JS targets, but the documentation mentions testing tools that log to the console. Using a similar approach might be possible.
As far as debugging in the browser goes, some guidelines on the first link above might help set up a relatively better debugging experience. You can also expand the object on your screenshot by clicking the arrow on the left side, which will display fields in a better format, but that only helps so much.