This kind of complaint applies to any method. Dimensional analysis just means that your data satisfies a certain kind of symmetry. A scaling symmetry in this case. It doesn't matter how long you define one meter, as long as you do it consistently. Hence any answer should not depend on the length of a meter. There are other symmetries, for example it does not matter where you put 0 on the temperature scale. Hence your answer should not depend on where you put that zero (unless that zero has special significance in your context). This is a kind of additive symmetry. Another additive symmetry is that it shouldn't matter which year you call year 0 (unless it has special significance, e.g. you are investigating the birth of jesus). In the same way your data can have any kind of symmetry, especially with multi-dimensional data you often get extra symmetries. For example it shouldn't matter in which direction you define north and east.

For any given problem, you should generally only use methods that obey the symmetries that your data has. This doesn't mean that PCA is invalid as a method, it's just only valid on a data where the scaling symmetry does not apply. Another example would be fitting a line through the origin for temperature data. That's invalid, because the result you get depends on where you define your zero (but as before, it might be valid if that zero has special significance in your context). Does that mean that fitting a line through the origin is invalid for any data set? No.

In other words, the same criticism could be applied to any given method. Just choose any symmetry that the method does not respect, and then declare it completely invalid. Hence we cannot dismiss a method outright purely based on this reasoning. For example PCA is perfectly valid on unitless data. What's even stranger is that the author does like neural neworks, which are certainly not dimensionally valid, heck they probably don't satisfy any real world symmetries. This is also a case where it can be OK to use a dimensionally inconsistent method. As long as it works, it works.