I can't say I have ever been mistaken for homeless, but I have had plenty of people offer me a place to stay for the night, food, drink, etc.
When I am driving, which unfortunately I do a lot of, if I see a touring cyclist I will often stop for them and ask if they want a cold water or need anything. I keep a cooler of bottled water in my car when I am driving as well as pretty good supply of snacks and other stuff. I have bought meals for others just as a means of paying it forward or backwards if you will. Never know when I might be in need.
I bought a coffee for a guy the other night at McD's he was a vet and looked a bit on the shabby side, turned out he was riding his bike from somewhere in FL to NJ working his way up the east coast. We sat and chatted for a few minutes over our coffees. Turns out he is actually retired military and draws retirement, lives very simply and prefers to travel by bike or walk everywhere he goes. Said it takes him about 3 months to make the trip. Apparently he has done it more than once. His rig looked a bit homeless, but was in good repair and well put together.
Aaron