Why is the boundary of friends-of-friends (FOF) halo corresponding to iso-density contour?
If you have enough particles to sample your field, then a given (local) density corresponds to the same mean distance between your particles. The FOF algorithm keeps linking particles to a halo until a particle is farther away than the linking length. That means that the halo boundary will be ~exactly on the border between regions of interparticle distances smaller and larger than the linking length, corresponding to a boundary between larger and smaller densities, respectively. That is, an iso-density surface.
In contrast, a halo finder using the spherical overdensity (SO) algorithm will always result in a spherical boundary, which will only be an iso-density surface in the case of a completely symmetrical halo.
This is the reason that FOF halo finders tend to include more filaments outside the region identified by SO halo finder. This may or may not be desired.