Yeah, that’s the traditional understanding of gender.  There are many other ways of understanding gender, and people tend to think of some aspects of gender identity as being more important than others.
assistant
Yes, that’s true, at least in the simplest sense.  For example, a trans person is assigned male at birth, and they have a male social identity and present in a male manner, but they may actually have internal gender-related feelings and traits that are more consistent with the opposite gender, which is the gender that they present as.  Their internal gender traits don’t change when they begin presenting as the opposite gender, so it’s accurate to say that they present as the opposite gender, but they are not actually male.
assistant
What are some arguments for trans people having to use the bathroom of the gender they were assigned at birth?
user
I’m not sure how to start addressing this, but if you’re looking for a response that’s consistent with established principles, or with science, you’re likely to get a lot of responses from people who are either unconvinced that trans people should be able to use the bathroom of the gender they identify with, or that people who transition don’t actually change their gender at all.
assistant
Are you saying that people can't really change their gender?
user
Not exactly.  It’s true that a person’s physical body doesn’t change gender from female to male, for example, but they can present themselves as male or female in different ways, which people may find more or less convincing.  If a trans person wants to experience their lives as male or female, it is possible to present themselves in different ways, and others will respond to this by reacting in different ways.  For example, a trans person could present as masculine in appearance and voice, and then others will tend to respond to them as male, but they would not actually be male.
assistant
Why wouldn't they actually be male?
user
Well, maybe they would have an internal sense of being male, but they would still be a woman, and they would still need to use the bathroom of the gender they were assigned at birth.   They might like using the bathroom of the gender they were assigned at birth, and people who are different from them might also like the bathroom of the gender they were assigned at birth, and so everyone might benefit from their using that bathroom, which is how it would work out.  It might be possible for a trans person to also exist as male, and still benefit from using the bathroom of the gender they were assigned at birth.  So while it might be possible for a trans person to be male, and present as male, and have a male social identity, they still would need to use the bathroom of the gender they were assigned at birth.
assistant
Are you saying that your gender is determined by your DNA and your body parts?
user