Any macroscopic deformation of a filamentous bundle is necessarily accompanied by local sliding and/or stretching of the constituent filaments. Yet the nature of the sliding friction between two aligned filaments interacting through multiple contacts remains largely unexplored. Here, by directly measuring the sliding forces between two bundled F-actin filaments, we show that these frictional forces are unexpectedly large, scale logarithmically with sliding velocity as in solid-like friction, and exhibit complex dependence on the filaments' overlap length. We also show that a reduction of the frictional force by orders of magnitude, associated with a transition from solid-like friction to Stokes's drag, can be induced by coating F-actin with polymeric brushes. Furthermore, we observe similar transitions in filamentous microtubules and bacterial flagella. Our findings demonstrate how altering a filament's elasticity, structure and interactions can be used to engineer interfilament friction and thus tune the properties of fibrous composite materials.