The image was still kind of small but by the looks of it yes, they'd be the same as there is no tolerance on basic dimensions so when you stack them it's equivalent to having a dim back to the origin.
That said, there are still pro's and cons to each way, it just doesn't affect tolerances.
Basic dimensions are used to define theoretically exact (perfect) location, orientation or shape of a feature. Usually they have to be specified together with geometrical tolerances because it has to be somehow defined how much abovementioned characterics can be different from ideal.
Basic dimensions themselves have no directly applied tolerances, so in your case both examples mean the same (if I am guessing correctly what is on your sketch).
Stacked dimension often have the advantage of better capturing design intent, and being easier for people using the drawing as interface information etc. to work with. Certainly, when I was checking hole patterns, if they dimensioned the hole spacing it was a lot quicker to verify etc. Most of ASME Y14.5-1994 tends to show this style.
Going back to the datum all the time may be a bit better for inspection & even manufacturing but doesn't capture design intent the same way much of the time.
So, the customer may have a point, but from a finished item point of view it shouldn't make a difference.
Based on the information provided, X & Y or X & Z could be equivalent/identical callouts, and likely are. To be 100% sure I'd need to see more of the drawing. I can think of some unlikely scenarios where it would matter. E.g. if Y or Z is part of the definition of a centerplane datum, then omitting that dimension in favor of another style clearly won't work.
I would ask why the customer wants to change the scheme. In my experience, requests for changing between equivalent dimension structures have often been a sign of not understanding GD&T.
