Aw, that’s heartbreaking.
I would tell him I understand. That it’s awful to miss someone you loved, that I miss the dog too, and that what he is feeling is a very big and difficult emotion called “grief”.
I would ask him where he thinks the dog is and what he’s doing, maybe encourage drawing because that can help children process their feelings. If he has any fears or worries about his dog being lost or scared, emphasize that you believe he is safely resting. It’s true. Beware of saying “we can get a new dog”, or otherwise implying that it’s not a big deal. He needs connection in his sadness. If he were grieving for a sibling, everyone would understand his deep emotions.
I’d ask if there’s anything we can do in this world to honour the dog. What would he have wanted? For example, if the dog loved playing fetch, maybe we can buy a toy and give it to a neighborhood dog or the local shelter as a gift. Or we could go out at night and light a candle while thinking about the dog’s favourite things. A child I know once grieved for my dog by writing a letter and leaving it by a stream, the dog’s favourite spot. You can also read books about loss and grief.
There needs to be room for his grief, because it can actually be dangerous to push feelings like that away. I wouldn’t worry too much about the idea of reincarnation – this is just the logic of a child. His dog is gone, but he can’t imagine him being “dead” – who can, really? So he wants to follow in the dog’s footsteps, believing they can be reunited.