Coming from Firefox and being a platform-hopper myself, I too was looking for a cross-platform link sync. I asked ChatGPT Deep Research what the best open source, self-hostable, cross platform bookmark sync system is. It looks like Linkwarden is the answer.
It is popular (has the most Github stars, >10k, of all apps in the category), AGPL-3 licensed, and actively developed by a company which offers a hosting service for it. It has a Chrome or Firefox extension. It also supports both heirarchical (tree / folder style) links and tagging, as well as performing full-text indexing for search.
LinkAce, linkding, and Wallabag get honorable mentions.
More in the report: https://chatgpt.com/share/67d107e1-e9e0-8011-86c2-f76bf9ba4992