The "Common Tongue" of the Andals is, of course, the English language which book readers and TV viewers observe. Author
George R.R. Martin has stated that he is not a linguist like J.R.R. Tolkien was, and thus he never thought out all of the languages in his fantasy world to the extent Tolkien did. Infamously, Tolkien made it a point that characters in his books actually
do not speak in English:
The Lord of the Rings is presented as if it were a "lost Anglo-Saxon saga" that was rotting in a library until Tolkien "rediscovered it" (much as the actual Anglo-Saxon saga
Beowulf was lost for centuries and then rediscovered). Tolkien worked under the conceit that the "Common Speech" (or "Westron") which the major characters in his works speak isn't really English, he just "translated" it into English. For example, "
Frodo Baggins" is just a translation of the real Westron name, "
Maura Labingi", and "
Hobbit" is a translation of the real Westron term, "
Kuduk". This extends to the point that certain significant names, puns, or allusions would only fully make sense in "real Westron", and not in English.
Martin has made no such comments about the "Common Tongue" in his fantasy world, and while Tolkien's Middle-earth was intended to indeed be our world in a lost prehistoric age, Martin has stated that the fantasy world of
A Song of Ice and Fire is truly an alternate world with no connection to our own. Thus it would seem that the English used in it doesn't "represent" the "real Common Tongue" through universal translation for the sake of the reader (and TV viewer). The Common Tongue just happens to be exactly like English.
Note that, from examples of writing from both the books and the TV series, "the Common Tongue" is specifically American English, and thus follows American English grammar rules and spelling standards (though admittedly, sometimes grammar and word choice can be intentionally archaized to make it sound pseudo-medieval).