IMHO a very capable FOSS competitor for Obsidian is LogSeq (https://logseq.com/). I use Logseq since 6 month and love it. There also exists a Kanban Plugin for LogSeq, which I have not yet tried... https://github.com/hkgnp/logseq-kanban-plugin
Obsidian may not be open source, but it reflects a sane approach to product: small, focused team, profitable company, delivering useful stuff. Clean and tight. It's got a different set of abstractions at its core (document model vs block model) but if you're indifferent to that, Obsidian is where I'd start.
But from the home page I don't directly get whether this is backed by markdown flat files or some other system, do you know?