What you're feeling isn't paranoia, it's your instinct telling you something important. When someone says "I trust you if you let me check everything," they're actually saying the opposite: they don't trust you. Real trust works without needing to monitor, and if he thinks he needs access to your chats to feel secure, then the problem is his insecurity, not your lack of transparency.
What's worrying you is legitimate because privacy isn't the same as having secrets. You can have nothing to hide and still need a space that's just yours, you know? Your conversations with friends, private jokes, even things that don't make sense to share with a partner. If you give in now "because you have nothing to hide," this doesn't end here: next he'll ask for passwords, want to know where you are all the time, and so on. It's the typical control pattern that starts subtly.
Two years into the relationship, if there's still this much distrust, it's worth having a serious talk about what's going on. Did he do something that made him lose credibility? Or is he bringing insecurities from before? Because a healthy relationship is based on mutual trust without conditions, not on surveillance. If he keeps insisting on this, that would be a red flag for you.