In this video it shows the Isle of Wight and the area around it including Southampton and Portsmouth being under Roman control beginning during Julius Caesar's invasion of Britain. However my brief research into the history of the Isle of Wight, Southampton and Portsmouth confuses me as I see no mention of them owning it before the official Roman invasion. Did they actually own that area before the official invasion during Claudius' reign?
It's more accurate to say the video depicts Roman control of Hampshire, not the Isle of Wight per se. That island is too small to really be seen on such a video of the world map. This means that this video likely meant to depict the Atrebates kingdom established by Commius in Britain, which was known to control much of modern Hampshire and Sussex.
Numismatic evidence suggests that Commius was settled in Britain as a king who would owe his loyalty to Rome. This practice was widely used by Roman generals to assert their influence over territories beyond the frontier of the empire. Commius took up residence south of the Thames, exerting his power over Hampshire and Sussex.
Eaton, Jonathan Mark. An Archaeological History of Britain: Continuity and Change from Prehistory to the Present. Pen and Sword, 2014.
The video's depiction is, of course, quite wrong. Commius was an envoy of Caesar during his 55 and 54 BC invasions. He eventually switched sides during the uprising of the Gauls, but fled to Britain after being defeated in 51 BC. So, even if we count his Belgic kingdom as part of Rome, it definitely wasn't in existence during Caesar's invasions.