Root is highly asymmetrical
While each faction has technically the same goal (reach 30 victory points), and there are some universal ways of gaining victory points (destroying your opponents' cardboard, and crafting items), each faction has its own unique mechanics for playing and earning other victory points.
The Marquise de Cat earn victory points by building Sawmills, Recruiters, and Workshops. No other faction even has these buildings, let alone gets victory points for building them.
The Eyrie have a building of their own (Roosts), but they don't get victory points for building them - instead they get victory points based on the number of nests they own every turn.
The Woodland Alliance earns victory points for spreading Sympathy, which superficially resembles the Cat's method of earning points, but the Cats build using wood (which the other factions don't interact with at all) in areas they control, and the Woodland Alliance spreads Sympathy without needing control (or even board presence) by discarding supporters (an extra 'hand' of sorts, which the other factions don't have.
The Vagabond earns points for giving cards to the other players and completing quests. The other factions can't even do these things, much less get victory points for them.
The asymmetry of Root comes from the highly asymmetric mechanics of the factions, rather than an asymmetry in the factions' goals.
Root isn't perfectly balanced, but it's practically balanced
Perfect balance is nigh impossible to achieve. Even if every player has identical abilities, any game with sequential player turns will have at least a slightly unbalanced nature, meaning that the only games with the potential to be perfectly balanced are ones that involve simultaneous play, like Set or Bananagrams.
But a game is practically balanced if the advantage difference between the factions (or turn order positions) is smaller than advantages created by randomness, player skill, and play dynamics (i.e. who's being beaten on by all the other factions this time).
Root is practically balanced. The differences in overall strength of the factions is slight enough that it's not obvious who will have the advantage in any given setup, and vagarities of play have more of an effect on the winner than any inherent advantage that faction has.
I'm pretty sure that the Lizard Cult from the expansion is weaker than the other factions. Certainly nobody in my playgroup has managed to win with them, and I've heard that they're getting some power errata with the newest expansion. But the four base factions are extremely well balanced.