Are primary, secondary, and tertiary amines considered functional group or chain isomers? Consider $\ce{C3H7NH2}$.  Can one make $3$ different isomers with this molecular formula? My question is that are they functional group isomers or chain isomers? 

I have been told that they are functional group isomers but I intuitively think that they are chain isomers.
When we talk about chain isomers we mean isomers with different carbon skeleton.

If the carbon skeleton is changing as well as the functional group we report it as functional group isomerism.

For the given compound in your question, primary, secondary and tertiary amines will show functional group isomerism as they are forming different types of amines itself.