Amplification of the "common mode" means that if we tie the + and - inputs of the op-amp together, so that it's not doing its job (amplifying the difference), i

Amplification of the "common mode" means that if we tie the + and - inputs of the op-amp together, so that it's not doing its job (amplifying the difference), it's possible that the op-amp amplifies the voltage present on the combined + - node; the voltage between that and the ground reference. Ideally, the amplifier should not do that, but they do. The gain is not nearly as much as the differential gain, so I think the CMRR (common mode rejection ratio) is something like how many decibels down from the payload gain is this unwanted common mode gain. Common mode gain can cause or contribute to an output offset. But if the amplifier circuit is biased such that + and - are kept very close to zero, that should not be the case. We normally care about the common mode rejection because common mode noise makes the + and - inputs swing together, causing the amplifier to amplify noise, like RFI/EMI. In your current sense example, although we have a 10% error, it is a systematic error (a displacement), not 10% uncertainty in the value. It is an inaccuracy, not an imprecision. You're assuming that the amplifier just buffers the output of the current sense resistor, so that the +/- 0.05 V across the resistor is translated to the output of the amp via unity gain. We could configure the amplifier to amplify the signal, so that it becomes +/- 5V (or whatever). Then the Vos of 0.0005 will be a 0.1% inaccuracy.