'Up' in this form is almost always a space-filler. It doesn't indicate any actual direction.
I don't want to take up any more of your time
I've used up all the sellotape.
I've eaten up my dinner.
There are cases it is needed, and I would suspect the earlier examples have borrowed from these uses, as a generic 'sounds like, so use it like' until the original reason for the distinction has blurred.
I took up the slack in the rope.
I did the washing up.
This last is a particularly British usage. The act of cleaning the plates after dinner is "washing up", to distinguish from the washing of anything else - clothes, or yourself. The US seems to use 'washing up' to mean cleaning yourself. This provides a source of amusement for Brits when we hear such as "I'll just wash up before dinner", which raises comments such as "Why don't you wait until after the plates are dirty?"
There is a strong case that to "take up" is to fill, as a complete idea.
There is also the usage 'to begin to do' but this isn't the case here.
See Cambridge Dictionary - take up