The problem is what does it mean to be biologically a man? Is it a static thing defined at birth or is it a description of the living organism as it exists? (It's the latter.) For example, there are some animals that can change their sex naturally. We don't say they're just the one they're born as.

OK, so now humans. If biological sex is a description of the person as they exist currently, what does it mean for us? Chromosomes are a useful tool because they contain the code that tells our body how to develop, but the actual development is the part that matters, not the chromosomes. The chromosomes will dictate what hormones are produced, and the hormones are what actually control development. We can control what hormones are in the body, so we can hijack the process and change the actual development.

So, since biological sex is a description of the creature as they are, if we hijack the process of development to tell the body to develop according to given sex, that's what their biological sex should be called, right? The clownfish that was once a male that changes into a female is a female. We don't say it's a male just because it once was one.

I can't say whether it's offensive. I'm a cis man. The issue I do know is that it's used by transphobes to pretend like they know more than they do and harm trans people. For example, congress's anti-trans bathroom rule. Speaker Johnson said: "All single-sex facilities in the Capitol and House Office Buildings — such as restrooms, changing rooms, and locker rooms — are reserved for individuals of that biological sex" He's using the term as a weapon, not as a descriptive tool.

Where it's most important is for doctors. My understanding is assigned sex at birth and medical records and understanding who the person is now is the useful information. They do need to know sex assigned at birth, and they also need to know if they're on HRT or have had other procedures. They have to treat trans people differently than their sex at birth because biologically they are different.