Practice having the purpose of your conversation be to understand the other person, not to have the other person understand you.
a few potential reasons (there's overlap here):
1. you and he have a cycle/pattern of behavior that is easy to fall into -- and you don't have that with parents/sibs
2. it is more emotionally charged with him, perhaps because the issues are closer to your fears.
3. you have had many years to learn how to have conversations with parents/sibs. the dynamic is, of course, different.
4. the nature of a partner relationship is far more intimate than parent/sibs, and this carries a lot with it.
If at all You are serious about Him,
perhaps You should Learn Him more, try to be rational in approach and stop being egocentric which most females tend to become knowingly and unknowingly with the passage of time.
Ask more questions; including those which challenge your beliefs as well as his.
Develop the confidence to have your beliefs challenged and the strength to change what you believe.
Be willing to have different beliefs about things and appreciate that the difference can be interesting.
Remember always that there is only one person you can change, and that is yourself.
It's odd that you can't talk at all without getting into a conflict. It would be worthwhile to figure out which one of you is turning conversations in that direction. It seems like you should be able to talk about at least 90% of subjects without any disagreements.
I would try to figure out which one of you is steering conversations toward arguments and how that is being done. It's hard to do that in the moment, so I would try to remember and write down what each of you said later, after it's over, and go back and analyze where the conversation started going wrong.
If you can identify the "trigger," you may be able to avoid it in the future.
Yes, you don't want to avoid conflict, but you also don't want to bring up difficult subjects every time you talk. It's a balance. Should you let something that bothers you slide and never mention it? Of course not, but you also don't want to choose the middle of a romantic dinner as the time to bring up the subject again. A study of successful relationships/marriages found that many of them had arguments and even shouting, but that they also had "good times" outnumbering bad times by 10 to 1. If you find that your ratio is a lot closer to 1 to 1, or you even have more negative interactions than positive ones, you need to learn to practice some strategic avoidance of hot-button issues -- at least for a while.
Much of my answer has been adequately covered in previous answers above. There is no one more important, there is no one more interesting than one's self. That is to say, let your partner talk about themselves and ask questions until you have a (more) complete understanding. Perhaps then they would ask about yourself and you would feel better having your partner know where you stand on things. If not, you might not want to make this person a life partner and end it amicably. And remember, if you don't want to know, don't ask.