If he's easily stressed and always has something to vent about, then I sympathise.
However, the way you've described your actions is not one of strong empathy. Sure, it could be worse. You could give him a verbal kick up the bum and say 'tough tits' (and some people need that), but to outline what I mean -
there is always a solution, sounds like you’re working with (colleague name) to deal with it, you’ve always overcome these challenges and I’m sure you’re the first person to go through it / have a nasty client
This first phrase comes across as you pompously shutting him down followed by minimising his problem by saying he's not the first person to go through it. He knows this. No one is the first person to go through anything, so it's pointing out the obvious and generally doesn't make people feel better. If you had lost your purse/wallet/mobile, or to take a more example, if your pet died, would you honestly feel better if someone told you that 'you weren't the first person' to experience that?? Like honestly, what are they meant to do with that information??
Sure, if they're afraid that they've made a mistake and are afraid of getting into trouble with management over it, or they genuinely think they're alone, then it might be of comfort, but other than that, no. It's just minimising their emotions.
Sometimes, when someone has had a bad day, they just need to vent, they need to feel listened to and they just want someone to say 'I'm sorry to hear that' followed by a hug maybe. That's it.
I’m allowed to talk about it though
Of course he's allowed to talk about it. What he doesn't like is feeling like you are shutting him down, like he's not allowed to talk about what's bothering him.
I recently feel like the cause of problems when I’ve genuinely done everything to try and make life easy when there’s stress at work for him
The things that you think are making life easy for him might not be that perceptable to him, or even apprciated. What he might need is the above response which I've outlined.
I sympathise if you feel exasperated with him though - some people are neurotic and they do get huffy at the smallest things which can make life really miserable.Worse, many people like this show no interest in helping themselves. They almost want to wallow in their own distress.