If you think about it, no two things are ever the same. It appears to me that you interpret "same" as identical. In the human-scale real world (i.e., not in, e.g., math and not on the atomic scale), two things may be approximately the same, but they are never exactly the same, i.e., never identical. Think of two current coins from the same country and of the same denomination: they are equivalent in terms of buying power but they normally have different amounts of wear or different years of issue or even different designs. These coins are not equivalent to a coin collector, for whom design, year of issue and condition are very relevant. I can construct many similar examples.
The dictionary definitions you quote are, as in so many other cases, incomplete. To make all dictionary definitions complete would expand the sizes of dictionaries beyond all reasonable bounds. E.g., the unabridged Oxford English Dictionary, when printed in a normal font, is more than 25 thick volumes (26, if memory serves).