[osaariki]

This is wholly untrue. The guarantees HE gives for security of encrypted data is exactly the same as any traditional symmetric or asymmetric crypto system. HE is indeed malleable by design, but doing computation on homomorphically encrypted data gives no information on the contents of the ciphertexts.

>This is more like anonymization (effectively a one-way hash) than encryption. If you encrypt 2 different values with the same algorithm and key, you will get the same ciphertext which reveals information about the original value (i.e., that they are the same).

This is a misunderstanding. Homomorphic encryption schemes are randomized encryptions schemes, which means even if you encrypt the same value twice with the same encryption key you get two different ciphertexts (both look like "random data"). They just have the property that they will decrypt to the same value. This is possible because the space of ciphertexts is much larger than the space of plaintexts.