You are essentially right in speculating that polymers are more stable, i.e. less reactive compared to the monomers that it is constituted from. I think a part of the reason is because of the changes from the existence of a double bond in the monomer compared to single bonds in the polymer, which makes it less reactive because of the decrease in electron density in single bonds compared to double bond, which reduces its reactivity with electronegative elements such as fluorine, oxygen, chlorine.
However, this generalisation can also be partly attributed to the consideration that chemical properties such as reactivity with other chemical substances is dependent on the molecular structure and bonds that a substance has, and because there are a variety of possible polymerisation processes, such as addition and condensation polymerisation, where depending on the conditions of the polymerisation, can result in formation of cross-links, branched and straight chain polymers, where the addition of inhibitors and other chemical substances can all come to have a significant influence on the final properties of the polymer compared to the monomer, resulting in it being generally much more stable and less reactive.