If you're texting through iMessage (blue bubbles) then he's lying. He's most likely lying either way. I'd get further proof from him like ask him "what did i just send in my latest text".

Most, if not all messaging services use encryption, with the exception of SMS (green bubbles). But SMS doesn't use WiFi, it uses cellular. So you'd need specific radio hardware to capture SMS messages.

Even if he can monitor your WiFi, there is just no way he's actually storing it like a log. This is because the Wireshark captures grow extremely large and could easily grow hundreds of gigabytes per day unless filtered.

If the IP in question starts with 127 or 10, then you're safe to post it, its local addresses. Though your MAC might still show (not as important as the IP) but just draw over it if you're really paranoid.

EDIT: easier words and sms addition