You can't pick your ethinicity either, it just exists.

If by ethnicity you mean genetics, then no, you can't change that (not yet anyway). But if you mean outward appearance of ethnicity, then yes, there are plenty of ways to adjust it that are widely employed. Everything from skin whitening cream to tanning to bleaching to plastic surgery to whatever Rachel Dolezal did. A lot of this stuff is very widespread in many countries, e.g. it's very common for Asians to get eyelid surgery to look more white.

People would change it if they could, same as height and any number of other immutables that people are discriminated on.

People are already getting leg surgeries to adjust height. It doesn't seem worth it to me, and there's lots of potential negative side effects, but this is something that is actually happening.

And I don't even need to explain all the surgeries people do to add or remove fatness.

Its just the technology doesn't exist yet. Although some feature of ethnicity can currently be changed.

People make choices about culture all the time, it’s in the realm of self determination.