Funky can often mean cool or stylish in a strange or unconventional way, however in this context I believe it's the old man referring to himself as appearing strange even to himself, perhaps from changing so drastically throughout his life that he almost doesn't recognise this person whom he has become

Funky also means cool or trendy or something like that. It comes from Funk, the type of music. It's a positive adjective and suggests the person/thing is modern and stylish.

Smelly and funky are not two interchangeable words. Not unless the word "smelly" has become a slang term for funky. But I haven't heard of smelly being used like that. The song you can interpret however you want. Nobody will agree with what others suggest and it is not a good song from which to learn or study English.

"Funky" originally did mean "smelly" in a literal sense--often specifically to body odor--but that was a long time ago. It has become generalized to mean a variety of things, including a music style. What they have in common is that they are not "nice" or respectable or traditional. In context here, it doesn't mean "smelly." It means _not_ looking "good." It means that he is disgusted by what he sees in the mirror. I find from a web search that "The song talks about Rivers Cuomo’s time at Harvard University after having corrective surgery on his legs nd his desire to go back to his old lifestyle, recording songs, touring with his band and making sweet love all night." So in this case it refers to someone who was young and healthy, and now is disabled because of a difficult recovery from surgery. He looks into the mirror, and is disgusted and depressed by what he sees. So "funky," here, does not mean smelly. The usage note in the American Heritage Dictionary says "When asked which words in the English language are the most difficult to define precisely, a lexicographer would surely mention 'funky.'"

‘Funky’ can refer to smell, but doesn’t mean ‘smelly’. It means ‘has an irregular smell that isn’t positive’. This room smells funky. ‘Smelly’ means ‘smells bad’ Here, the meaning isn’t 100% obvious but he might have perceived himself as cool and stylish, but what he sees in the mirror doesn’t agree with that belief.