So let me get this straight. HH acts like a cyclist. Another cyclist tells HH that he is not a car. HH concurs.
HH, I'm glad you didn't read me stories as a kid.
Joking aside. True, you're not a car. Also, you're not a driver, you're a rider. I know they rhyme and there are other similarities, but ultimately riders and drivers are different entities. You are also correct in stating that you have the same rights. However, in the context provided, this doesn't quite work. Stopping behind a line of traffic isn't a right. To have accurately reflected matters, you should really have tempered this by also stating that riders have the same responsibilities.
Some of the other differences between riders and drivers that you should take into consideration are our very different set of vulnerabilities, capabilities and opportunities.
Out of interest, what do you genuinely believe that the other cyclist took away from your exchange and how will that make them a better rider?