Can it be useful to standardize the observations over the entire dataset even when the units are the same?

Normally when we standardize the different data points to compare the different data on the same scale.

But does it also make sense to standardize data points even when the different data points are already on the same scale?

For example, suppose that I have a dataset {x, y, z}, where x, y, and z are in the identical unit. Does it make sense to standardize these three data (i.e. subtract the sample mean of {x, y, z} from each x, y, and z, and then divide the quantities by the sample standard deviation of {x, y, z})?

If there are cases where such standardization can be useful, what are the examples of those cases?

Thank you,
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Ambassador to the humans
I mean I guess it depends on what you're talking about.

1) how many feet tall are you?
2) how many feet can you long jump?
3) how many feet can you sprint in 30 seconds?

Those are all on the same scale (feet) but they aren't really directly comparable.