[secondcoming]

In theory it can:

How should a Transparency & Consent String be stored?

In version 1 of the TCF Specifications the consent string was specified to be stored as either a 1st party cookie for service-specific consent or a 3rd party cookie for global consent. In version 2 of the TCF Specifications, the storage mechanism used for service-specific TC Strings is up to a CMP, including any non-cookie storage mechanism. However, global TC Strings must still be stored as cookies under the consensu.org domain.[0]

Pretty much no website uses it.

[0] https://github.com/InteractiveAdvertisingBureau/GDPR-Transpa...

[mhils]

In practice, some CMPs used to share positive consent across websites, but did not share negative consent. So if they tricked you into accepting once, they keep it; if you refuse, they keep annoying you. My understanding is that watchdogs pushed back, which is why the whole sharing thing isn't as prominent anymore.