It's totally normal for a baby not to sleep all night until a year or more. My baby started getting up just once around 9 months and still does. It's okay. It's a stage and he will eventually sleep. Just because some other babies start sleeping through the night at 2 months or because your ped says he "can" doesn't mean he is personally ready to do it. Every baby is different.
You should not keep him up all day. Getting overtired is not good for him; he NEEDS sleep -- at least 2 naps, probably 3 naps -- a day to be healthy and well-rested.
Be patient. Take naps and go to bed early if you can so you can be rested even if you have to get up several times a night. Have a good bedtime routine. Play white noise while he sleeps. Make sure you feed him 'til he's full during the day so that he hopefully won't be as hungry at night. But realize that he's perfectly normal ... most babies don't sleep all night at this age -- my baby was getting up at least every 2 hours at that age! It was rough, but we got through it and she now only wakes once in the night, and that's just fine. I can do once! She'll give up that waking soon enough, too.
By the way, at 6 months I believe the longest period of wakefulness they're supposed to have is 2.5 hours. Follow his cues, when he starts to get grumpy, rub his eyes or nose, put his head down on you, yawn, etc. you should put him down for a nap. It's counterintuitive, but "sleep begets sleep", meaning that a well-rested baby tends to sleep better than an overtired one.