How do you know he is not hungry?
6 months old is a growth spurt time....
You said you feed him at 9, 2 or 3 am, and 6 or 7 am. Is that per HIS waking/hunger... or a feeding 'schedule' you have for him???
If you are using a feeding 'schedule'... then that may be the problem. He is not feeding when HE needs to. If you try feeding him when he wakes/needs it... then that may solve it.
I breastfed both my kids.... they woke at night, JUST like your baby.
They had GINORMOUS appetites. They woke every 2-3 hours. I nursed them at night if they woke.
I also had a lot of milk... so they were satisfied. Not hungry all the time.
I would also check your milk output.... are you producing enough?
Growth spurts and its ensuing wakings... also exist so that the Mom's milk production also increases, per baby's needs/growth. Intake.. has to keep pace WITH the baby.... and growth needs/spurts.
If not, they are hungry.
Also, maybe he is teething? 6 months that happens as well.
Wakings is also due to their MANY developmental/cognitive changes.... and hitting milestones. If he has hit some major changes in development recently and/or milestones... then this ALSO tweaks sleep.
AND... he sounds REAL over-tired. Over-tired babies... actually do NOT sleep better, they have a harder time falling asleep and staying asleep.
Also, MANY babies, get over-stimulated REAL quick, and that does not help sleep either.
Babies, under 1 year old... will typically get real tired, after about 2 hours or less, after being awake.
When he does nap... is it at regular times? Or just whenever? Or only if he is nursing???
Maybe just try a pacifier??? Maybe he is real oral? My son was like that.
Maybe it will help him to self-soothe...
Your baby is normal though. BOTH my kids were like that as babies. But they did nap, well. But they woke at night.... a lot.
all the best,
Susan