I think it depends on how you look at things, though.

Scientifically speaking, there is evidence that your levels of dopamine affect your personality:
Dopamine linked to a personality trait and happiness | Cornell Chronicle

And I wouldn't be surprised if serotonin, GABA, etc. affect your personality, too.

Whether or not the effects of neurochemical levels cause noticeable enough changes in personality is subjective. We can obviously identify changes in personality via various tests; therefore, we know these changes exist when neurochemical levels are altered.

Now in my opinion, the amount by which your personality changes due to the neurochemical imbalances caused by bipolar disorder depends on you, the person. For some people, these changes are quite evident. Perhaps for you, the changes are so subtle that it feels like they don't exist... which is a fair assessment. I don't expect everyone to notice changes in their personality.

However, I will agree that mood greatly influences your behavior, but behavior and personality are two distinct entities. Though, I would argue that personality does affect behavior. An extreme case is antisocial personality disorder.