Here's why:
* "Boarding" usually implies getting on a larger vehicle like a plane, train, or boat.
* "Getting into" or "getting in" a cab is the more common phrase.
Here are some correct ways to say it:
* "I'm getting into a cab."
* "He got in the cab and drove off."
* "We're getting into a taxi."
"Boarding" could be used in a very specific context, such as if you're describing the process of getting into a cab specifically for a long journey or a tour, but it's not the standard way to say it.