@ma1985cg
Yes you can use Customer Attribute for unauthorized website, but you need to make sure that implementation is proper and you need to set 'authstate' to Authenticated.
Because in order to deliver Target experience based on Customer Attributes it is must to have.
You can refer below docs:
Set AuthState
Customer AttributesFor Target
Yes I agree with @pratheeparunraj , that you can implement Experience cloud ID/Eloqua ID as CRM ID to upload into Customer Attribute to target the users.
But how do you sync an ID with multiple Experience Cloud IDs since Experience Cloud ID is generated differently every time and stored as a separate CRM ID? So, functionally not advisable
Regards
Skand