I know P ~ V^2 assuming constant impedance. I am not arguing with you about the physics. "3dB" does not include any information about units. "3dB" is a ratio, it doesn't care what it is a ratio of. "3dB" is a purely quantitative concept. "3dB" can apply equally to elephants, libraries of congress, Volts, or kielbasas.
My post about "3dB" being conceptually equal to "200%" is correct because "3dB" has nothing inherently to do with power or voltage or anything except a *ratio*. Only if one adds units to it, can it *imply* a *ratio* of something.
"3dB" is the same thing as "2:1", "2", "200%", "2.0x10^0", "0x02", "2/1", "16/8", "two to one", "two" and "II". "2" does not mean "2 Volts" or "2 Watts" or "2 overly pedantic slashdot posts" or 2 of anything else unless you put units after it to indicate that it is supposed to mean *two of something*. "3dB" is in the same unitless boat as "2". I can say "I have 200% of the chickens I used to have" or "my chicken gain is 3dB" and either way, I will be saying the same thing, because "200%" is a ratio, (200:100), and so is "3dB", (2:1), and (200:100) == (2:1).
Any ratio can be expressed in deciBels; to do so, one takes one's ratio, takes the base-10 log of it, and multiplies that result by 10. This final result is the ratio in deciBels. *That* is how deciBels are defined, regardless of what their application is or what the ratio represents. Any definition of the deciBel where the ratio 2:1 equals anything other than 3dB either has the deciBel confused with something else or is confusing the definition of the deciBel with the definition of whatever they are trying to use them for.
The assertion that deciBels are a unit of signal strength is wrong unless, by "signal strength", you mean "signal-to-noise ratio." DeciBels can only quantify signal strength when they are used in conjunction with a reference signal strength and then they only describe signal strength as a ratio relative to the reference one. If you want to call deciBels units of something, they are a unit of *ratio*, and that is *all* they are. They are not inherently a unit of signal strength, sound pressure, voltage, power, or anything else. There is not a physics book in the known universe that disagrees with me about this if you read it carefully. When someone speaks of "signal strength" in "dBm", like other posts have pointed out, they mean a quantity of power or voltage relative to 1mW or 1mV respectively -- a ratio of mW to 1mW or a ratio of mV to 1mV. "dBm" != "dB".
Again, you are right that P ~ V^2. You are right that when you change the voltage by a ratio of 1.414:1, you get a change in power of 2:1. When the ratio of (power now):(power before) is 2:1, when before turned into now, the power went up by 3dB. When the ratio of (voltage now):(voltage before) is 1.414:1, when before turned into now, the voltage went up by (10*log10(1.414))dB.
If you want to indicate that P~V^2 using deciBels, you could say that a 3dB increase in voltage implies a 6dB increase in power, or that the ratio of the power ratio to the voltage ratio is 3dB. Anyone who simply states in a textbook, (textbooks are supposed to be clear and unambiguous,) that a 2:1 ratio comes out to an unqualified "6dB" when the ratio happens to represent voltage and then provides a formula to make it work out that way is suffering from cranio-rectal inversion. It always has been and always will be incorrect. If your textbook says otherwise, whomever wrote it should be soundly chastised after having their cranio-rectal inversion cured. :)
I realize that this post has exceeded standard allowable limits of pedantry, but if you will refer to my initial post, you will see that I summarized this problem by saying something about someone turning a voltage knob while reading a power meter and incorrectly claiming that a factor of 2 is 6dB. The problem is not that they are wrong about the physics, the problem is either that they don't really understand what "6dB" means or they haven't explained that they are turning a voltage knob but measuring power, thereby making their ratio a ratio of two different kinds of things where X units of one thing really does translate into X^2 units of the other.