Standing alone, yes, decibels are ambiguous. But decibels are as ambiguous as saying "times two" without referencing what you're doubling. There should always be a reference.
The gain listed in dB is referring to the input type. So...
What's your input?
I'm not sure I've seen something as ambiguous as a device called "amplifier" with its gain in dB and no other information. You'll usually have a voltage amplifier or a current amplifier; an amplifier with the input being voltage or current respectively. That is, if you have a voltage amplifier, then the input is voltage and the output is the voltage with gain. A voltage amplifier with 3dB gain will turn a 1V input into 1.4125V (0dBV to 3dBV).
If your input is dBm, then the input is power, and the output is power with gain. For your example, a device with a 0 dBm input and a +3dB gain, the output will with 3dBm. Similarly, if the input is 0dBW the output will be 3dBW.