The passage referenced is metaphorical. That which is being stored up or given away is not material possessions but emotions. People who bottle up their emotions are hard to please and people who deny their emotions (give them away) are hard to please.
From a literal perspective (where what is in question is material possessions), it is factually inaccurate to say that monks do not store up anything or give everything away. An individual monk may do this, but most monasteries and Wats where monks reside do maintain a store of supplies. At least that is my observation.
Monks do not generally give away their alms food nor do they store it up, instead they eat their alms food and are satisfied. Monks do not store up robes nor do they give robes away, instead they wear just their 3 robes and are satisfied with that. Monks do not store up medicine (for long anyway) nor do they give medicine away, instead they use medicine given when they are sick and are satisfied with that.
I think that is the point, although as Metaphor said, it isn’t entirely accurate, at least not nowadays. But wandering monks back in the day certainly wouldn’t be storing things up or giving them away.