How did Cambridge come to be considered a City despite not having a cathedral within its boundaries?

In an indirect way, it was in recognition of the university (it was the site of the only "ancient" university not in either a city or a royal burgh), however, city status in the UK has nothing to do with having a cathedral, a university, population or anything else like that. It's based on whether a town has been given a royal charter to call itself a city. Example: Guildford has a cathedral but it's not a city.

Having a cathedral is not related to city status. Neither is having a university, a swimming pool an airport or anything else. A city is created by royal decree at the whim of the monarch. Now it is usually done to mark a significant event such as the Queens diamond jubilee or her 80th birthday or some such event. Applications may be invited from towns wanting to be cities. Chelmsford, a cathedral town in Essex, was granted city status this year. Nothing has changed apart from the fact it is now called a city.

A city is defined by either having a cathedral or a university within it's boundaries.

You can also become a city if you have a university

I thought it went by population. I guess the traditions in England are a little different.