Homomorphic encryption means you can ask Apple "who is calling me" without Apple knowing who is calling you.
You hash your query and then send only the first X number of bits.
The server returns all results that hash up to that same first X number of bits.
The server doesn’t know exactly what number you were looking for, and you don’t have to download the entire database.
But in this case the server WOULD be able to figure out the set of possible phone numbers you were asking about. Because of the complexity of passwords the search space would be a lot larger.
So privacy wise this does seem better.
Though there is a valid argument that you're still leaking information (e.g. "Person X received a call at 21:05:43"), but I'm not sure how you could possibly make an API that avoided that given the time sensitive nature of identifying callers.
[Disclosure: I work on the team responsible for the feature]
For instance, if you often receive a call at the same time of day, that could be a detectable signal in the noise, unless the client then creates a lot of similar fake signals in the noise.