This is not a problem with you and him, this is a problem with you and your girlfriend. Your expectations of her son do not match hers. Does she mind cooking for him? Does she consider him a part of her caregiving circle still? The discussion you need to have is with her. If you'd like him to contribute to family expenses, ask her about this and how she feels about it. I don't think that is unreasonable. However, if she wants him to have access to food in the house, eat meals she cooks, I also think that is reasonable. I sincerely doubt his dream is to live at home for eternity. But you can discuss long range plans for his final departure. He's at that age in which a lot of kids are lost. Maybe offer up some ideas for him to have a better future. Something like trade school. Electricians and plumbers make a lot of money! Try to guide him but in terms of his living in the house with or without contribution is something to discuss with his mother. SHE is the one making his rules and has determined how things go in the house for him. Not him. He's just doing what he's always done. So, talk to your GF about it but do so without being judgmental. good luck
If the 20-year-old's job is keeping him away from the house except for 2-3 days a week, then I would presume that he is earning a very good salary/wage. And I believe that asking him to pay a modest rent as well as "chipping in" for food is not that big a deal. Requiring him to help cook and clean the house and do some of the grocery shopping should be part of the ground rules for his living in your home - maybe a "roommate contract" such as that used on the "Big Bang Theory" is called for in this instance!