No. A cache hit writes nothing — no counter, no log, no database connection. That is both a privacy property and the reason it is fast.
The Dashboard's Pages Not Stored panel keeps, for a sample of pages the cache refused, the reason, the page's address without its query string, and the name of the cookie that stopped it. Never a cookie's value, an IP address or anything else about who visited. The Clearing Log keeps the user ID of the person signed in when the whole cache was cleared only when they work on the site (a user who can edit posts), never a visitor's, a customer's or a subscriber's.
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