He's lying to you about something. The question is, it is something you can live with?
My first thought was, "He's married" just like the other writer. But it could also be that he's a giant slob. I dated a guy like that for three months -- never saw his place because he has problems with clutter. Then he went back to his ex-girlfriend. Good riddance. Or your guy has a kid, or he's gay/bi. The problem is that you don't know and he's not telling you.
So, if you, a woman (presumably), would be questioned if you went over, how about sending over a male friend, someone you trust to tell you the truth? If he truly has a male cousin he lives with, he wouldn't mind having a male friend of yours come over to check out the place. I would ask for that. Ask in person and watch his face. If he's not lying to you, he can say, "That's a great idea." But if the answer is anything other than "yes, let's do that" then you know he's lying.
Now, you could do some research. You could google his address and see who owns the place. (You do at least know where he lives, right? If not, you could look at his driver's license while he's in the shower at your place.) If he's the co-owner with a woman, then you know he's married. Or you could drive past the place. Or you could have a friend "deliver" flowers to him and see if he's alone or really living with another man (and what if that's his gay lover?). But look at the level of deceit that's involved with all of this.
Here's a different question: are you meeting his friends and work colleagues? Are you seeing him at different dates and times? Presumably, if you meet people who know him, someone's going to tip you off if he's married (or not, I met a LDR guy's work colleagues and no one told me he had a local girlfriend). If there are certain days/times you don't see him, like he says he always had basketball on Wednesday nights, then he's got a kid and that's the time he has custody.
Basically, you don't trust him. I think you know he's not being truthful to you. So I would say to him, "Look, I don't buy the 'male cousin would rat me out to the family' story, what's going on" and see what he says. But a liar is a liar, and if he can't be truthful with you after five months, then there will always be lies coming out of his mouth. It's not like he's going to end up being a good guy except for one minor lie.
Looking at your other questions, don't bother buying him the coat until you have this question answered. And this question about where he lives has been bothering you for at least three months, and yet you keep going. Why is that? Time for a therapist, I think.
FWIW, the guy who's a keeper knows by five months that he wants to be with you. He's already planning the ring and the proposal. This guy isn't. He's got some giant secret and he can't be honest with you. You can keep trying to make this work, but the question is Why?
Five months in he should have been showing you off to his family a long time ago.
Watch the comedy movie "Real Genius". "Why won't you introduce me to your family. Are you embarrassed of me"? "No, them!". "That's okay. Let's go meet them". "Sure". I get the idea of being embarrassed of or by family. but that's something to get over.
Five months in he has decided you are NOT a keeper. React accordingly.