Your question seems to be related to how information is stored and used.
Basically YES information is stored by ANY server/site or system (and browsers).
This is a common set of how ANY cms/website stores and handles information.
ANY website/cms will store information depending on level information is needed.
A visitor storage will normally store ip, browser, display size, OS, time,etc.
These (by cookie or session) will allow to interact between a 'vistor' and the site visited and 'rule' the way communication is build upon.(page access, prev, next , images, downloads etc..)
This kind of storage is 'hidden' and normally not public viewable. ( a system admin, could access this, but on request (legal) provide such info. (normally called 1st party).
A true 'user' (internal access) with possibilities to interact 'inside' a system, needs different policies and storage of information, to uphold such a communication acceptance (eg needs to remember who and where/why did change/access, is allowed, and so on).
Although very brief answer (others may of course reply also) it will not inform you on the exact policies as these are to be seen, in appliance and confirmity and applicable to the law's of each country.
A cms will NOT provide you with (example: EU GDPR/VGA rules and settings) but may provide you with help on this, being pre-build forms, fields, page(s). THIS (alike GDPR but in general, all information storage (access,viewability etc) is YOUR responsibilty.
Systems MAY provide you with further help on this by using extension/plugins (ex. cookie related).
Any plugin/extension that connects to the'outside' world has it's own set of rules (eg 3rd party). Some may provide access to policies in use (linkage), else you need to inform users yourself. (will not be an automatic process).
Next level would be the kind your website/cms is related to, eg medical/government/hobby/private etc.
All need different set of rules to comply with the regulations needed/maintained by law/usage and so on.
Although this is mere a universal answer (and a very short one as is)., the moment you start up your browser (including private browser view/vpn), it sends out data, which will be used one way or another (storage/usability).
I can go on and on (pages with texts) but a more accesible example would be : imagine your site being a spiders web, you are in the heart of it and each branch is a connection your site makes to the outer world.Googles stores, your isp stores, the site stores, a plugin connects, CDN connections, libraries etc.. Sorry..one may not be anonymous,
Your job as a web(master/owner) makes policies your are bound to or comply with, not a/the system.
The system only can help you in certain ways.
Need more info? please reply or pinpoint your request in possibilty..