I think this may be a combination of personality and still adjusting to new roles. If you just began staying home a month ago, you had the holidays in there that might have been unusual, and so you might not have a whole month to really get used to things.
I go through this a bit every summer when I go from teaching full time to working part-time/not at all. My kids are used to my undivided attention for most of the time when I get home, because that's what they get when I'm working. But I can't do that 24/7.
Some of it may be your son's personality too - some children simply need more attention/interaction than others. My nephew was incapable of playing by himself. From a very young age. He's an extreme extrovert and really gets energy by being with people. My kids are somewhere in the middle. They need interaction, but they also need some time to be off doing their own thing. (Thankfully for me, the introvert who needs time to myself to recharge.)
Things that help:
-Having a routine or a schedule so they know what to expect. It can be flexible, but knowing what's coming really helps the flow of the day. A visual schedule with pictures is often very effective with a younger child. When I'm home, we get up, have breakfast, the kids get a little TV, we go out to the park, we come home for lunch, they're expected to play on their own for a bit while I rest/do e-mail, we run errands if necessary, play together (indoors or out), fix dinner, play after dinner, pajama time, snack time, clean up time, bed time.
-The kitchen timer. I'll often set the timer and say "I'll play school with you until the timer beeps, and then I have to start dinner/do a load of laundry/read the newspaper.." That gives a definite end to the activity and allows me to give my full attention to the activity, knowing I'm not stuck being a student for the next 3 hours.
I also use the timer to let me kids know when I'll be ready to play with them again (though that's not as necessary as they get older), and to warn for transitions (we don't transition well in our house).
-If I need more than 20-30 minutes of down time, either I have to leave the house or dh and the kids do. We had some extreme parental favoritism with ds from about age 15 months until 5 or so. Age 2-3 was the hardest because he didn't want daddy to do anything with/for him if I was home. (Ironically, he was home with dad exclusively from 5 mos to 2 1/2, then in daycare 3x a week from 2 1/2 on, and home with dad 2x a week. When I wasn't around, they did fine together. But once I was home, forget it.)
So, dh and ds (and later with dd) developed a routine of outings every other Saturday. That gives me 2-3 hours at home to recharge, read books and putter around the house. It also helps cement their relationship.