Traffic law (in the US): The law considers the bicycle, first and foremost, to be a mode of transport, and sees a need to regulate the flow of bikes the way it regulates the flow of automobiles. This dates back to the dawn of automobiles, if not before (though in some states it took decades for the law to spell things out as it does). This makes sense, since bicycles can attain speeds well in excess of a normal pedestrian and their speed creates hazards for both them and pedestrians if they ride on sidewalks or otherwise behave as pedestrians.
Besides, if bicycles were pedestrians we'd not be allowed on roads, in the flow of traffic. This would place a major constraint on cycling few of us would want to see.
I believe most states have an exclusion of sorts for children on bikes or maybe even for adults riding at low speed, allowing them to use (most) pedestrian paths. But this is an exception to the general rule.
There is a problem that a significant fraction of the general public does not tend to view bicycles as vehicles, but this is an education problem, and I'd definitely not want to confuse the issue by raising the above question "in public".
(I am reminded, however, of "Big Lip Louie", a guy who lived in rural Louisville, KY 50 years ago when I was a kid. He was, at the time, maybe 30-40, and didn't drive, but walked everywhere. He was tall -- well over 6 feet -- with long legs, and could walk at speeds I'd guess were in excess of 10 MPH. It was fairly common to encounter him walking along a rural road, pretty much right down the center of the road. He was more of a "vehicle" than a pedestrian.)