In Modern Hebrew there's a difference between עצם=bone and עצם=object: a bone is a feminine noun and an object is masculine. Since it's העצם הזה and not העצם הזאת, you know it's masculine and therefore it's "this object" and not "this bone".
Except that people will also use זה for feminine objects as well, and עצם is much more commonly used to refer to a bone than object (according to my friend), making more reasonable to assume it's a bone rather than an object.
Not in this case. "זה עצם" could be "this is a bone" (colloquially), but "העצם הזה" can only mean "this object".
Oh wow thanks for clearing that up, that means I will have to be a lot more careful with my "זה"s and "זאת"s
Many thanks for all your explanations, I find them very helpful. But still - how are we sure that these words are absolutely unrelated? Both tree and bone are something strong or solid which can hold other things... Couldn't they stem from the same origin?
Well, for such question you have to compare the other semitic languages too, and Ugaritic, Phoenician and Arabic, Ethiopian and Akkadian too have variations of עץ, all meaning "tree" or "wood" with two radicals, but the root-family עצם is attested in verbal and nominal forms in Phoenician, Ugaritic and Arabic. There is no usual development or any indication, that these words have interacted.