If one says "pick up the cleaning", does this mean "go to the laundry and pick up the clothes that have already been washed and dried"? I didn't find such a meaning of "the cleaning" in online dictionaries.
If one says "pick up the cleaning", does this mean "go to the laundry and pick up the clothes that have already been washed and dried"?
The phrase itself, is not unheard of - and is comparable in popularity (in literature) to similar phrases. google n-gram
However, the most appropriate use of "pick up the cleaning" is as a short-form of "pick up the dry-cleaning". Rather than regular machine-washed laundry.
That said, especially if you are going to collect your clothes from a laundrette (rather than your own washing machine) - it would be extremely unlikely to cause anybody much confusion over what you mean.
As such, I see no realistic reason that using this phrase could cause you any trouble or misunderstanding in a real-life situation.
That said, while it works as a specific phrase - because you are going to another location to collect "cleaning" (which forces it to be a collectable/holdable item) - relying on this phrasing may cause you to make mistakes when extending it to other situations.
I need to do the cleaning
Would give the idea you were going to do house cleaning (sweeping, dusting, tidying etc.) and not anything relating to laundry.
With that in mind, I'd recommend you use one of these common ways to refer to your personal (not dry-cleaned) laundry:
I need to pick up the washing.
I need to pick up the laundry.
Which will avoid any ambiguity, and can be correctly extended to other forms ("I need to do the laundry", "I wish I didn't have so much laundry" etc.)
You could call it "the washing" (https://dictionary.cambridge.org/dictionary/english/washing) or "the dry cleaning" (https://dictionary.cambridge.org/dictionary/english/dry-cleaning) but not just "the cleaning."