Ever since the novel quantum Hall effect in bilayer graphene was discovered, and explained by a Berry phase of 2 pi [K. S. Novoselov et al., Nat. Phys. 2, 177 (2006)], it has been widely accepted that the low-energy electronic wave function in this system is described by a nontrivial Berry phase of 2 pi, different from the zero phase of a conventional two-dimensional electron gas. Here, we show that (i) the relevant Berry phase for bilayer graphene is not different from that for a conventional two-dimensional electron gas (as expected, given that Berry phase is only meaningful modulo 2 pi), and (ii) what is actually observed in the quantum Hall measurements is not the absolute value of the Berry phase but the pseudospin winding number.