Why Do Most of The Papers Use the Frobenius Norm for Denoising?
The Frobenius Norm has multiple equivalent definitions – the useful for error measure is probably this one:

$$\left\|M\right\|_\mathrm F = \sqrt{\sum_{p\in M}\left\lvert p\right\rvert^2}$$

That's a root square over all pixels.

Root mean squares are very useful cost functions, as they describe the power of a signal.