Is it possible to overcook peas as one could overcook a steak or pasta, or are they like mushrooms in that they cannot be overcooked?
Yes! Fresh/frozen green peas can really easily be overcooked. They'll lose the bright green, getting more dull and a bit darker. They'll also get mushy. If you've had both nicely cooked fresh/frozen peas and canned peas, this should be pretty familiar to you.
If you mean something dried, more like beans, then sure, you can still overcook. The color is already a bit of a lost cause, but they'll still get mushy with extended cooking.