And yes what you said is exactly right; you're allowed to fingerprint a unique user and track data with that fingerprint as the sole unique identifier without any PII legislation (GDPR, CCPA, etc.) compliance issues. You just cannot store any information that allows linking PII data to that fingerprint in either direction. In other words, attribution to a random UUID that just happens to represent an anonymous user is not an issue.
Circling back to the original comment; there is no (good) argument against cookies if you're basically doing exactly what cookies are doing. Umami using it as a USP is, at best, a little odd.