FT illustrates two examples of First-price Auction, Ex 6.5 (continuous) and Ex 6.6 (discrete).
In Ex 6.5, they establish the uniqueness of the equilibrium when the reservation price is higher then the lowest valuation, and the equilibrium is symmetric. Their basic argument is that when the above condition holds, Lipschitz continuity is also satisfied, which implies the unique solution to the differential equation derived by FOC.
In the footnote, they mentioned the multiplicity of the solution of the war of attrition.

Standard results on the uniqueness of solution to differential equations require Lipschitz continuity. The war of attrition of example 6.3 is not Lipschitz continuous at s=0, which is why it is possible for the system represented in equation 6.3 to have multiple solutions.