"My mother goes to the city with the subway."
Translation:Mijn moeder gaat met de metro naar de stad.
Yes, Canada has Subway, but it also has Mr. Sub (https://en.wikipedia.org/wiki/Mr._Sub). I think most of the major Canadian cities have many U.S. companies operating in them.
Canada also has Kernels Popcorn - https://www.kernelspopcorn.com/Default.aspx
As a Torontonian, I would probably say either
"My mother goes to the city on the subway"
"My mother goes to the city by subway"
"Take the subway" is also used (e.g., "How are you getting there?" "I'm taking the subway.")
I wouldn't ever say "I'm going with the subway".
So, yes, the English translation here sounds unnatural to me as well.