It is possible for equivalent bonds to have different convexity. Remember, convexity is calculated from duration. Duration is a measure of time / sensitivity (regular / modified). Both convexity and duration relate to time. Convexity is the true sensitive of future cashflows whereas duration is the approximated. If you really want to get in the weeds, you discount each periods cashflow by not the yield of the bond ,but each tenor's yield along a curve. Intuitively when rates fall or rise the bond with more time left to maturity will react more because you are discounting more future cash flows. The key part here is maturity.
It is not impossible to have two bonds with equivalent price/yield. A simple example is a sovereign yield curve that is completely flat. Each bond would have the same price and yield , but the convexity would differ hence DV01s would differ.
Now to answer your question about market behavior. Well it depends on the market participants views. Convexity is not always viewed as attractive. From a trading/investing prospective you might spread the lower convexity against the higher convexity. Each handles a different regime. If rates are rising, you could long the low convexity and short the high convexity or in falling rate environments, short the low convexity and long the high convexity. The main goal of these strategies is to trade the shape of the yield curve removing the parallel shifts.
Please let me know if that answers your question.