I've attached the responses from Kaspersky.
To summarise:
Official browser support includes dedicating resources to testing and maintenance for that specific implementation. They would not elaborate on the scope, but I would assume they look for bugs/vulnerabilites etc.
Popularity of a web browser determines whether it is worth dedicating resources to.
While the KIS Chrome and Edge extension are the same (and therefore compatible with most other chromium browsers), non-supported browsers are untested. In the worse case, the KIS extension may cause either itself, the web browser or the Kaspersky application to not function correctly.
They are currently compiling user feedback for Brave browser. If you want to see Brave supported, I recommend filing a ticket using the support link below. The more feedback they receive, the more likely it will be implemented.