That's actually the Declaration of Independence that says that. And it means what it says.
Not all of us pursue our own happiness by owning things. For many of us, pursuing happiness means being able to study certain things, or being able to practice and execute certain skills. What the founders were talking about were fundamental rights that we all have as human beings, because they are required aspects of our being fully human. Without life, our 'humanness' is irrelevant. Without freedom, we are existing, but not really living. And without the ability to pursue that which makes life meaningful to us, we cannot become fully ourselves. The founders did not see property is the essential factor for individual human expression. They saw the need to pursue our dreams as that essential factor for full human expression. And so that's exactly how they worded it. They did also state that these three were rights "among others". They implied that there are other fundamental unalienable rights, and if you want to label one of those as the right to "own things", I doubt many would disagree with you.