Does the Collector's pass grant level skips?

The boost lands the moment you activate the handbook, not gradually as you play. Ten handbook levels arrive at once, which means the first ten tiers of battle pass rewards are already yours before you have caught anything. If you plan to buy the Collector's tier at all, buying it early in a season is the version of this that costs you nothing, because the head start is applied from wherever your handbook sits when you press activate.

Treat the level skip as one part of a bundle rather than the whole purchase. The same $20 tier comes with ten extra Aniimo storage boxes, which is the difference between releasing creatures you were undecided about and holding on to them. It also includes exclusive evolution materials tied to branching creature evolution, so some evolution paths are gated behind owning the handbook rather than behind grinding for them.

The monthly pass is a separate thing and works on the opposite principle. It pays out 6,000 companion experience points spread across 30 days, so you have to keep logging in to collect all of it. The Collector's Companion Handbook front-loads. The monthly pass drips. They are not substitutes for each other.

Not settled: The ten levels are a one-time grant at activation, so it is not settled what happens if you activate deep into a season with fewer than ten tiers of rewards left ahead of you.