Creator of theq extension here, that is an optional permissions (may ask). It never asks to access to all sites, it only optionally let's you add support for additional sites such as different Invidious instances. These are arbitrary domains, so it needs permission to ask for any arbitrary website.

Support for invidious is not actually done yet, so right now, it never asks for permission for any new sites anyway

The client side data is used to store your submissions to always show your preferred title, even if there is another submission on the server. You can disable this option in the settings

Access your data for all websites

It's required to access iframes.

Without having investigated the plug in itself, going by just what you posted: That's not likely to be a nefarious request, the plug-in needs to store data on your computer. Dealing with large number of titles it makes sense that it would cache things on your computer to minimise overtaxing their servers, to serve you more or less the same data over and over.

Maybe it's for embedded videos? idk