Do British lawmakers make laws for and govern British Columbia ?

British Columbia is a province in Canada which is it's own independent nation so no.  The British Commonwealth is a lose collection of countries formerly part of the British Empire but now independent.  It is no different than the UN or NATO when it comes to being governed.  For the most part they still have, at present, Queen Elizabeth II as their titular Head of State although her authority is strictly limited to basically ceremonial duties and she holds citizenship in these countries so, in Canada, she is the Queen of Canada.  All laws and regulations governing the country are made and approved by the Parliament in that country or, in the case of Canada, the Canadian Parliament which is comprised of elected representatives from all regions of the country.

The area was established as a British colony in 1843.  The name distinguishes it from the U.S. area still occasionally known as Columbia for the Columbia R. The river has its source in British Columbia, but flows down to become part of the border between the U.S. states of Washington & Oregon.    

British Columbia became a Canadian province in 1871.  It's governed by its own provincial laws, the Criminal Code of Canada, & various statutes of Canada.

There's no such thing as the "British Commonwealth" anymore.  It's been known since 1949 as the Commonwealth of Nations.  It's 54 member nations include the U.K. & other free & independent nations, most of which were once British colonies.   

No, obviously not. 

by that logic, Mexico makes laws for new mexico?

No, that;s a stupid thought process.

Are the Virgin Islands run by virgins?  I heard that joke in grade 5 ffs

"Im curious why a place that’s part of the British commonwealth wouldn’t be ruled by the British?"

We're used to people with no clue, we just chuckle and keep walking.

To answer your question, because life doesn't work that way anymore.  "Evolution" seems to pass you by also.

"The Commonwealth is a voluntary association of 54 independent and equal countries."

https://thecommonwealth.org/

Note the words voluntary and equal   :)

Just because a country is part of the British Commonwealth, it doesn't mean it is still governed by Britain.

No.  BC is in Canada, and has been for a very long time.