They are not dogs

Guardian dogs are installed many shrines for protecting the shrines against what comes in form outside. Guardian dogs were originally installed in Buddhist temple but after syncretization of Shinto and Buddhism, guardian dogs were also installed in shrines, and have continued to this day. On the other hand, the role of guardian at Buddhist temples changed from guardian dogs to Nio, and guardian dogs were removed from almost Buddhist temples.
Many Japanese people call this pair of dogs "Komainu", but in fact, one with horn and its mouth closed is called "Komainu" and another one with its mouth open is called "Lion". Komainu and lion are not actually dogs, but imaginary creature and lion. Komainu and lion represent "Aun" same as Nio. Lion with its mouth open sounds "A" and Komainu with its mouth closed sounds "Un", which signify the beginning and end of all things.
Due to the separation of Shinto and Buddhism in the early Meiji period, there were instructions to remove guardian dogs, which originally had their roots in Buddhism, from shrines, but many shrines left them as they were. This is probably because the two majestic and adorable statues were cherished by the local people, and there was a strong desire to preserve them.