Yes, inherently noisy, and this can vary a lot between cartridges and arms even on the same phono stage. Usually anything low level and uniform like you're talking would be within the norm. You can try and reduce if it matters to you but could spend an awful lot of time for very little to gain. Except when it is one channel like you said. In that case you should be able to get the noisy one down to the same level as the other- unless there is something wrong somewhere.
That is what we're trying to figure out. That is why you want to try the phono stage somewhere else. If you get the same noises, it's the Allnic. If not, keep looking.
You picked up on the main idea, phono stages have a tremendous amount of gain. Things that would never be audible anywhere else can be unlivable with all the gain. Worth it in the end, even if it doesn't seem that way sometimes.