Uninstall chrome and use Firefox, and it works for me with full cookie sharing. There is even an "open in Firefox" in the hamburger drop-down which transfers the page to the browser app without reloading.
Last time I tried Firefox on my Android phone it showed weird drawing artifacts (half/quarter of the page is full black) on pretty must all websites, so thanks for your suggestion but I'd stick with an actually working browser.
In any case, I don't need to bother about whether any "cookie sharing" works or not if I only use one browser, do I?