In one area of the garden in the house I bought couple of months back, we have mulch and recently we are seeing the following. Not sure if this is a mushroom or not. My current guess is that the mulch is very old and needs to be replaced. But I would like to know more about why this is growing.
Definitively a mushroom, a cup fungus of the pezizaceae family. (Precise identification is difficult over the internet.)
What you see are the fruiting bodies - not unlike the apples on a tree. The fungus itself lives in the mulch / soil and is saprobiontic, which means it digests the wood chips of your mulch. This is perfectly normal and part of the breaking down of organic matter. You wrote yourself that your mulch is pretty "old".
I'll confirm that the identification of Peziza fungus is accurate in Stephie's answer, and it is a saphrophytic fungus, meaning it merely digests dead material, link below (if you scroll down) shows one or two which are more like yours to look at. These are commonly known as cup fungi. Now that it's present, even if you replace your bark chips with new, it's likely the mycelium will still be in the area, and you may get fruiting bodies like these appearing again.