The votes themselves are the federated action.
If you fetch an old post, your instance will not see the previous voters. After that, whenever a user votes the instance will get the message "User X@instance upvoted/downvoted post Y" and the vote will be added to the database with the voter's user ID and counted.
This has a practical function. If you don't keep a list specifying who voted for what, it would be much easier to fake votes from one instance to another by simply communicating the message "Downvote post Y". With the current method it is still possible to create a lot of fake accounts and mass-vote, but at least you can get a better insight when looking at the database if the votes are associated with accounts with no activity from a single instance.
There are some federated platforms that will show who likes / dislikes something. I know that friendica used to do this - I have not checked if it still does. So it is not only admins who can see this, this is is basically open information in the fediverse.
Yeah. You can see the cookies that are stored by a site by right-clicking on the site, going to "inspect", and the clicking "Storage". By default, the only cookie that Lemmy has is an jwt cookie used to authenticate your user.
You are not asked for a phone number to be here. Providing an e-mail is often optional and even discouraged by some instances. When you want to send a private message through the site you get a message discouraging from doing that and encouraging to try to use an encrypted chat application instead, such as matrix.
The original Lemmy instance (lemmy.ml) is a community for FOSS and Privacy enthusiasts. What is asks from a user and what it does with the data is what it needs to be functional. Lemmy lets you take any proactive step that you would like to take to protect your privacy - use a VPN or Tor, use safe passwords, use a unique identity, and don't provide any personal information. There are no built-in features to block you or discourage you from doing that. Lemmy never asks for your location, nor does it keep any logs of what content you visit, nor does it try to run any analytics on you. But even if that is not enough for you, the fediverse doesn't lock you out, you can set up an instance or even create a new program to interact and communicate only precisely what you want to communicate via activity pub.