Doctors have unions [1] ("among actively practicing physicians, approximately 70,000 currently belong to a union, representing 8% of physicians" [2]), lawyers have essentially guilds. If your wealth is closer to Brin’s than the median, than you’re an outlier whom I would not expect to need nor value a union (congrats on your luck). Unions are for the median, not the very wealthy and lucky [3].
The median annual wage for physicians and surgeons in the US was $239,200 in 2023. In May 2023, the median annual wage for lawyers in the U.S. was $145,760, with the lowest 10% earning less than $69,760 and the highest 10% earning more than $239,200. Stats shamelessly stolen from the US BLS website.
I'm also a lot closer to a lawyer or a doctor than I am to a plumber. And very likely on the Sergey Brin side of the distribution.