The grammaticality of "that don't impress me much"

The intentional misuse of don't is a form of code switching (or code mixing). The form is extremely characteristic of working-class southeastern Americans ("southerners"), who are also the primary audience for American country music.

What is most interesting about the song is that Shania Twain is Canadian — and that is where the code switching begins. It is a deliberate error made in attempt to establish authenticity and to better connect with her music's intended audience.

The use of don't instead of doesn't is an error in standard English, of the sort you were probably taught. However, this sort of error is characteristic of many non-standard, rustic dialects, and country music of the sort that Shania Twain sings is known for using these dialectical features as part of the conceit of being rural, Western [1], and unpretentious.

[1] As in the American West, i.e. cowboys and Indians.