Is the continuous communication of a male cousin with his female cousin, with whom he was raised, considered a betrayal in Islamic law, even if the conversation does not include obscenities, given that his wife is aware of it?

Your cousin is considered a stranger (non-mahram) in Islamic law, and your upbringing together does not change this ruling. It is not permissible for a woman to speak with a non-mahram man except out of necessity and while observing the Islamic guidelines, such as not speaking in a soft or alluring tone. The wife's consent or knowledge does not affect this ruling. Betrayal means that a person is trusted but does not give good advice. It has also been mentioned in the Quran in the sense of betrayal of faith, as in Allah's saying about the wives of Noah and Lot: "so they betrayed them," meaning in faith. It suffices to say that this action is forbidden and displeases Allah the Almighty.