Is ionisation of acids and bases in water is a decomposition reaction?

No it is not.

A decomposition
reaction is one in which the products on the right side of the chemical equation are
different than the reactant on the right side of the chemical equation. There is a
chemical difference between the reactants and products. In addition, a key factor in
recognizing a decomposition reaction is that there are more products than
reactants.

An example of a decomposition reaction is the
breakdown of water into hydrogen and oxygen.  2 HOH -->  2 H2  +
O2.

Another example is the breakdown of potassium chlorate
into potassium chloride and oxygen.   2 KClO3  -->  2 KCl  + 3
O2

In the ionization process, a compound is added to water
and the water separates the positive and negative ions in the compound.  This is a
physical separation and when the water is removed you still have the same compound you
started with.