Why do the elements of the same group have similar properties ?

We should be careful when defining "properties" - chemical
properties are probably the ones that an instructor would be taking into consideration.
By contrast, elements of the same group have very different numbers of protons, masses,
spectral lines, and abundances in the universe, so we cannot necessarily equate groups
with overall similarity among elements.

Chemical properties
generally refer to how we predict an element will react with other matter. Valence
electrons are the primary determinor of chemical reactions, and so we would expect atoms
with the same number of valence electrons to undergo the same kinds of reactions, such
as bonding with element X but not with Y, or releasing an electron in the presence of A
but not B. 

The connection here is that groups on the
periodic table are organized according to valence electrons; all elements in a group
have the same number of valence electrons. However, even here we have to be cautious
with generalizations; just because Hydrogen and Lithium have one valence electron
doesn't mean those electrons are the same, and will react the same. For example, the
higher the element number, the farther away the electron is. This will result in similar
reactions, but the strengths of those reactions will be considerably
different.