It's either the remnants of a slime mold or fungal mycelium. Neither is a concern as they are both digesting or feeding off of the mulch. Harmless and completely common.

Could also be actinomycete, looks like fungus but actually a bacteria, but conclusion is the same, harmless and normal décomposer.

This is composting (decomposition) in action. It is what is supposed to happen. Bacteria (which we usually can't see) eat the soft, nitrogen-rich parts of the plant, and then leave the rest for fungus (like this) to finish off the tough woody parts over the course of several years. If you don't want to see it, mix it up with a rake every few weeks to distribute it in the mulch. But, ultimately, it is good for the plants - it will produce compost and humus from the woody mulch over time.

I agree -- it looks like a good thing, a decomposer doing the needful to your mulch. I wish my mulch would decompose that well, but it's hard in my desert climate.