I am a chemical engineer whose career is based on analyzing gas mixtures, so I am quite confident when I answer this question.
Non condensible gases that can be considered roughly ideal (nitrogen, oxygen, argon, helium, hydrogen, neon, etc.) will completely mix very, very quickly. Their molecular velocities are high enough so that, on average, they are bouncing back and forth in that tank literally hundreds of times a second. Not only is this theoretically true, but definitely is supported by my personal observations. At work we have NIST traceable gas standard mixtures in cylinders, and they have certified contents stating the concentration of each component. One had a label stating it was produced in 1985 and probably hadn't moved in 10 years. I analyzed it by cross-referencing it to another, newer NIST-traceable gas cylinder using a mass spectrometer (the most accurate method possible for analyzing gases). The concentration was exactly the same as what was on the label, within experimental error. Actually, the simple fact that NIST will certify a primary standard gas mixture in the first place is evidence that it does not stratify.
Now, highly non-ideal gases, such as water vapor, freons, sulfur dioxide, etc. will tend to stick to the walls of the cylinder and could possibly stratify or have concentration gradients within the cylinder. This is due to their affinity for each other or for the metal of the cylinder. However, these gases are either not present in breathing mixtures (as in the case of sulfur dioxide), or their concentration is very low and relatively unimportant (water vapor).
As others have stated, the reason you stir other mixtures, such as sugar in kool-aid, is because you are trying to transfer mass from one phase to another! You are trying to make solid sugar dissolve into water. This happens more effectively when the concentration gradient is greatest. Thus you mix it up, otherwise the water "layer" right next to the sugar crystal saturates and mass transfer is slowed greatly. But in a gas cylinder, everything is in the same phase (and the most mobile phase!) so this isn't an issue.
As for those who have tested your mixture before and after shaking, I think it's due to temperature variations or random error in measurement. I will admit that I am not familiar with commercial oxygen analysers used in the scuba industry, so can't fully speculate why the error is occuring. My first suggestion would be to take 25 measurements of the exact same mixture and determine your 95% confidence interval. Do this by taking the standard deviation (use excel formula) and multiply it by 1.96. This number is the error in measurement, plus or minus. You now have a range of your analyser where you can be 95% certain it is right (provided there is no systematic errors!).
Maybe after I get my nitrox cert I will analyze a cylinder with one of the mass spec systems!