Consider a hypothetical API that returns the same value for 95% of users and a unique value for each of the remaining 5% of users. If we use the entropy definition of a privacy budget, calls to this API may fall within budget, but still uniquely identify 5% of users. If we use a k-anonymity approach to this API we would either need to disable it entirely (since a single call is sufficient), force the users who would return an identify value to return the same value as the 95% of users (which may lead to breakage), or only allow a whitelist of values to be returned (which bundles all of the otherwise unique users in a single group). Have you thought through how to handle such APIs?

As a real-world example of an API that may fall into this category, consider the ability of a script to retrieve a device's local IP address from the the RTCPeerConnection.localDescription property or the RTCPeerConnection.onicecandidate event handler. For most users this will return something low in the private network range, but for a small percentage of users this may return a globally unique IP address (such as those behind a university NAT). For the latter group, that's all that's required to track them across sites.