Is There Any Filtering Technique with Element Wise Product Only Instead of Convolution?
Classic filteration is indeed done using convolution.
Though I have seen broader definition of filtering as shaping the signal in its frequency domain which can be done in many other methods as well.

Of course you can create meaningful operations using element wise operations and even specifically multiplication.

Think of the case you have a noise with the following properties:


It is non negative.
It is signal dependent.


Then operation like thresholding would make sense and it is applied using element wise multiplication. This case is very similar to $ {L}_{1} $ Denoising.

Pay attention that in this case the filter depends on the data.