non native speaker here. when someone calls another person boy and the other person responds by saying "don't call me "boy" ". Does that mean the first person was calling the other person "boy" to imply that he/she is wiser (or smarter or has more experience in life etc.) and the other person rejects that by saying "Don't call me boy". I heard this in this video at 3:13 to 3:30. https://youtu.be/WyGq6cjcc3Q
In the given context, Alex Jones is trying to belittle John Oliver by saying him "boy". It is derogatory, if you call a fully grown-up man a "boy".
an immature male
separate the men from the boys
Also, it is an offensive1 word for a male servant.
1: Check Entry 2 in Merriam Webster