Sharingan literally means Copy Wheel Eye. The Eye of Insight and the Eye of Hypnotism are the two abilities of the Sharingan.
The Eye of Insight has different abilities in itself. One of the abilities, lets the user copy any jutsu and utilize with maximum achievable perfection.
The user is able to copy almost any jutsu they see, memorising ninjutsu, genjutsu, and taijutsu with near-perfect accuracy. They can then either perform that jutsu or modify it to suit their needs, as when Sasuke Uchiha bases his Lion Combo on Rock Lee's Front Lotus. Again, Sharingan users need the prerequisite abilities before they can mimic a jutsu they have seen, and for that reason they cannot, for example, reproduce kekkei genkai abilities they do not have or nature transformations they haven't learned to perform.
Therefore, every Sharingan user is capable of using the Eye of Insight's ability of copying other jutsu, provided, their Sharingan are matured enough. It is not, therefore, exclusive to Kakashi. Every Sharingan user has this ability. Although, Kakashi is the only non-Uchiha able to perform it successfully.