Replace cookie-universal-nuxt with a more generic library

[cnotv]

Description

Some logic has also been removed from a removal attempt of cookie-universal-nuxt (#11272).
As this creates several issues, we want to replace it with cookie-universal (found by @rak-phillip).

Logic replacement to be discussed/handled with #11392

[rak-phillip]

@cnotv can you expand on the issues that were observed with cookies-universal-nuxt so that I can better understand how to address this issue?

The underlying cookie-universal and others all appear to utilize a similar pattern when working with cookies (e.g. cookies.get(), cookies.set())¹²³

Footnotes

1. https://github.com/microcipcip/cookie-universal/tree/master/packages/cookie-universal#api ↩

2. https://github.com/js-cookie/js-cookie?tab=readme-ov-file#basic-usage ↩

3. https://github.com/bendotcodes/cookies/tree/main/packages/universal-cookie#browser-example ↩

[cnotv]

At the current state, I just found out that this library is set to keep cookies data updated into the global.
As you said, it should not be more than set/get.

[rak-phillip]

@cnotv and I had an out of band discussion on this. The existing approach in #11354 should satisfy our need to remove nuxt dependencies, but we might want to follow-up in a separate PR to address using the inject pattern with cookies. The implementation in #11354 has been tested and confirmed to be working in Vue3, so a larger refactor away from the inject pattern shouldn't be required for the near-term.

[cnotv]

I rephrase this issue to match to actual final implementation.