A truism in photography is that a full frame camera produces a more narrow DOF than does a smaller sensor camera such as APSC.
THIS IS NOT TRUE...
The DOF at the same subject to image ratio is identical despite the size of the sensor!
The truism that the larger sensor produces the more narrow DOF comes from the the fact that when shooting a subject with a larger sensor camera, we will normally shoot from a closer distance resulting in a different subject to image ratio that includes a more narrow DOF.
Shot with the same focal length lens at the same f/stop and the identical distance, the DOF between the larger format and the smaller format will be identical.
That said, even at the same subject to image ratio, when the subject only fills a portion of the frame, you will include more of the surrounding area and, UNLESS YOUR SUBJECT IS A FLAT FIELD, you will include more OOF area within the frame giving the impression that the FF sensor has a more narrow DOF than the crop sensor.
Pardon the pedantry - but a better word would be "falsehood" rather than "truism":
https://literaryterms.net/truism/
or: "Part of photographic lore is that ... "
Yes, but there are other circumstances in which the DOF will not be the same. The best summary I have found is this.Shot with the same focal length lens at the same f/stop and the identical distance, the DOF between the larger format and the smaller format will be identical.
Specifically OFF the topic of Macro and Close-up:
I think that there is much confusion around the reasoning behind comments like - "If you're interested in attaining really shallow depth of Field, then building a kit comprising 135 Format Cameras and really fast Lenses is the way to go."
And I think that confusion leads to less specific and inaccurate statements such as: "a full frame camera produces a more narrow DOF than does a smaller sensor camera such as APSC."
WW
I like vague because the purely technical definitions only deal with that aspect of photography. The physiological and psychological aspects are just as important and are not as easy to quantify.
Explaining why "warm" tones are the result of lower colour temperature illuminants or chromatic adaptation s are examples of the psychological and physiological aspects respectively.
Actually the answer seems to be yes, in many cases.
The people that tend to care are the ones that print and the discussions there are often ProPhoto RGB vs Adobe RGB and both the ink set and paper get pulled into the discussion as well.
Frankly most people I speak with don't even know if their computer screen is sRGB compliant. I have all kinds of Apple fans telling me how good faithful the colours on their iPhones and iPads are when I tell them that those devices are not colour managed. They also busily tell me that the P3 colour space is superior, even though the gamut has yellows and reds that cannot be reproduced by typical print processes photographers use and doesn't hit the greens and blues that Adobe RGB ( and photo inkjet printers) does.
Along the same line... Although I have a 60mm macro lens which is capable of a 2:1 image ratio (the image is 2x the size of the original). I really seldom use it anymore. My 28-75mm f/2.8 Tamron lens is capable of a 1:2 image ratio (image size is half that of the original) and since this lens lives on my Sony cameras, it is expedient for me to use it for most of my close up photography.