The grammatical purpose of pronouns is to keep from having to repeat proper nouns every time you reference them. Which would be tedious.
No, in everyday interactions, a person's gender pronouns are not based on their biological sex. They might be assigned to a person based on our assumptions about their sex; assumptions made based mostly on outward gender presentation. Most people have little direct access to the biological sex of other people so they infer sex from outward cues like clothing, hairstyle, and (less reliably) certain secondary sex characteristics. We assume a person has a female sex if they outwardly present in a conventionally feminine manner, etc.
Using gender pronouns consistent with another person's wishes is about being a decent human being who does not desire to cause unnecessary harm to others. It's really that simple: caring for other people.
I'd like this question ("Why is external validation required for internal feelings?") to be regularly asked about cisgender and heterosexual people who seem to require an ENORMOUS and ONGOING amount of social and cultural validation in order to function...
***EDIT: You’re obviously sea-lioning, so I’m not responding to any more of your disingenuous responses to my writing. Go troll elsewhere.***