I've always avoided using the common phrase "the exact same" because it sounds incorrect to me (unless perhaps a comma were inserted thus: "the exact, same".) Shouldn't "the exact same" be "exactly the same"?
In British English, yes. "The exact same" sounds (to my British ears, at least) like an American phrase.
According to these Google Ngrams, both American and British English use exactly the same more than the exact same. Here is the usage in American English:
And here it is in British English:
Despite its usage, the exact same is considered informal (but is not deemed incorrect) by this site at Washington State University:
However, there is a long discussion of the phrase which writes that:
Proponents of the phrase note, however:
Both phrases are redundant, and the exact same can be considered correct or not depending on which style guide one subscribes to. An American English grammar guide specifically mentions that the phrase is mostly standard, and a British English guide notes that there are similar phrases. So use depends on context: in formal writing, avoid it, but in anything else the exact same can be (again, it depends on who you follow) acceptable.