This affects cookies sent by server-side code only, so this is something that concerns Esri. These warnings are temporary and will go away with the stable Chrome 80 release, because at that point any deprecated cookies will simply be blocked. In the meantime, if you're debugging and these warnings annoy you, you can disable them using chrome://flags ("Cookie deprecation messages").
Should client side developers be concerned? That shouldn't affect your app at all, as far as I can tell, EXCEPT for one situation: if you're developing a Chrome extension that uses chrome.cookies. If your extension's logic relies on monitoring a cookie that becomes deprecated, then you have cause for concern, but there's nothing you can do unless you're also the server-side developer (or have the developer's ear!). Then, the Stack Overflow link given above is pertinent.