Well... yes. There are analytic chemistry techniques used to measure the concentration of phenols, which are the primary contributors to a "smoky" flavor. You could test samples taken from the meat at different depths to measure penetration. You'd need a reasonably well-outfitted chemistry lab to do a good job of this.

Honestly, though, objective measurements are not the right way to go about this. There's a reason food scientists do both chemical and organoleptic tests. Objective chemical measurements are always going to be an imperfect approximation of subjective experience. If the color of the test strip tells you something's not very smoky, and your tongue tells you it is, which are you going to believe?

If what you're looking for is a way to reliably compare the results of multiple attempts over time, just take small samples and freeze them, then compare them [using your tongue] once you're back at max taste.

I know of no such scale, however you could use a Subjective Organoleptic approach, which is how the Scoville chili heat scale worked (although they use a chemistry based approach for the most part now). My understanding of it is that you dilute food and test how much of it you need to detect the quality you are looking for, like sweet, acidity or smokiness. Strong flavors will be detectable much more diluted than weak ones, so by testing multiple foods you can build up a picture of the subjective strength of each on that characteristic.

My BBQing colleagues use the cut-test - a ring of smoke penetration can be seen (in pork and chicken at least, I don't know about darker meats) in the meat that has some bearing on how heavily the meat has been smoked. I don't know if this is standard practice or just something they came up with (Edit: turns out it is something BBQ judges look for).

It seems that this is related to how long the meat was smoked, how hot the smoker and probably how the meat was pre-treated (brined?, marinaded?, rubbed?).

In response to @phil's comment, I have been to the source of all BBQ knowledge, amazingribs.com, where they combine science and BBQ and had a look for smoke rings: It turns out that the smoke ring is actually linked to Myoglobin - a pink protein in the meat, and its exposure to heat, nitric oxide and carbon monoxide, NOT smoke (particles in suspension in the air) per se, but gasses made by combustion of wood or charcoal in the presence of air. It only happens at temps below 170 F (~77 C). So it doesn't directly measure how much smoking has gone on, but rather how long it has been in the presence of CO and NO at relatively low temperatures - often in direct relation to how long it has been smoking.

So - TLDR, the "smoke ring" is not directly related to smoke, but rather to the presence of gasses found in high abundance in smoke.

Use measured amounts of ingredients like smoke salt or liquid smoke.

If your current methods of introducing the smoky flavor to a dish can't be easily measured, you could consider using alternative methods to do so that could be. For example, you could measure out an amount of an additive like liquid smoke or smoke salt that you could add to the dish. That way, every time you use the additive, you'll be able to ensure that the same amount of smoky flavor is added if you use the same amount of the additive, and since you're not using actual smoke, you'll be able to taste the food to judge its flavor without having your ability to taste the smoky flavor drowned out by the smoking process.