A blanket that looks tired after a month is the most common complaint in this category, and it has nothing to do with how it was treated.
What is actually happening
Polyester fleece is a knitted fabric brushed to raise a pile. Brushing leaves short fibre ends near the surface.
Rubbing — against a sofa, a coat, the inside of a washing machine — works those ends out and tangles them into balls. That is a pill. It is mechanical, and it starts on day one.
What anti-pill finishing does
A properly finished fleece has been treated so far fewer loose ends are available to migrate, and the surface is set so the ones that do are held down.
It is a real process with a real cost, which is why cheap fleece skips it and still uses the word.
The number to ask for
Pilling resistance is graded on a standard test, typically one to five, where five is essentially unchanged after the test cycles.
Ask for it in writing. A supplier who has specified the fabric will know it; one who has not will change the subject to how soft it feels.
What you can do afterwards
Wash rarely, cold, on a gentle cycle, and never with anything abrasive like zips or towels. Skip fabric softener, which mats the pile.
That will slow pilling on any fleece. It will not rescue one that was never finished properly.