1950 children's science fiction story

Some of the details don't quite match. But, this reminds me a little bit of the 1960 Robert Silverberg novel Lost Race of Mars

Points that match:

I once read a 1950s children's science fiction story set on Mars.

Partial check: Story is set on Mars. Book published in 1960

It was about a family that had just moved there,

Check. The story is about Sally and Jim, ages 11 and 12 or thereabouts ("Jim, "six and a half" Martian years, and Sally, six" Martian years) along with their parents who move to Mars.

and befriended a colonial kid. They lived under a dome,

Check. Though many of the Martian children do not originally like Jim and Sally.

and adopted a Martian animal with red, white, and blue hair, which they named "Yank."

Partial check. They adopt a Martian animal as a pet. However, they call him "Mitten" for "Mars Kitten".

Book review

This is Richard M. Elam's 1953 'Young Visitor to Mars'. The family is from Earth, they live in a domed house, the teen-age hero makes friends with a Martian colonist his own age and they have a red-white-and-blue pet named 'Yank'. It is available on Project Gutenberg (here) and there are cheap copies on places like ebay.