Dutchess recently had some clear drainage when she had a viral/bacterial infection, and she was sneezing a lot and had a bit of a runny nose. The vet told me that clear drainage is better than yucky drainage, if its gunky or goopy or discolored then there is something going on. They did go through some allergy questions with me, so it may be that your bun does have an allergy. Since he’s only two months old, it could be something in your home that is causing it since he just moved in. I would -not- recommend using any sort of q-tip or tool to remove the drainage, I used my finger to swipe down Dutchess’s cheek to remove it rather than swiping it back towards her eye, be mindful of your nail so you don’t scratch. If it is something you don’t want to make it worse by spreading it. It doesn’t sound like he is in a lot of pain since he is eating and acting normally. Bunnies do have a normal drainage for their eyes, so once you make sure he is okay watch so you know how much is average for him, so in the future you know if it changes. Some light colored rabbits have yellow on the corner of their eyes from their natural drainage, and that is nothing to be alarmed about. Bunnies also get sleep crust which is fine to remove, I remove Dutchess’ every day and she seems to enjoy it.
Keep us updated after your vet check 