Similarities between Co and Ni Elements $\ce{Co}$ and $\ce{Ni}$ share many similar/same properties and usually if one of them is an example of some group (e.g., ferromagnetism, same possible oxidation states, packing fraction and coordination number in lattice, melting and boiling point,  similar properties of their alloys, etc.), the other is too.

I understand that it may be due to similar sizes and atomic numbers, but many other elements have these properties in common. Can you give any other example of such a pair and why only few of these exist, or why this relationship between  $\ce{Co}$ and $\ce{Ni}$ special?
Cobalt and Iron ions are the only ones to produce colored complexes when mixed with thiocyanate ion $\ce{SCN^-}$. If you mix up a dilute acidic solution of Iron(III) Chloride $\ce{FeCl_3}$ and a similar solution of ammonium thiocyanate $\ce{NH_4CNS}$, a extremely dark red solution is obtained, due to a complex containing the ions $\ce{Fe^{3+}}$ and $\ce{SCN^-}$. If you replace $\ce{FeCl_3}$ by a solution of Cobalt Chloride $\ce{CoCl_2}$, you obtain a blue color due to a complex containing $\ce{Co^{2+}}$ and $\ce{SCN^-}$. No other metallic ions are known to produce such colored complexes with thiocyanate ions, at least among the usual ions studied in chemical analysis courses. The blue Cobalt complex has also the property of being soluble in pentanol, where it can be extracted if the initial solution is colored.

Any way it should be mentioned that :

1) The reaction with Iron is extremely sensitive. The red color may still be seen if the concentration of the complex is 1 - 2 ppm. The blue complex made with Cobalt ions is only visible at much higher concentrations. 

2) If this reaction is made with a Cobalt solution containing Iron as an impurity, the solution will turn red, as the blue color of Cobalt will be hidden. In this situation, a couple of milligrams of Potassium fluoride $\ce{KF}$ should be added to the solution. This will produce a colorless complex of $\ce{FeF_6^{3-}}$ and the blue color of the cobalt complex is restored.