How common is it to have a funeral with the coffin open? Unless the friend had been in touch recently, he probably wouldn't recognize him after time's ravages anyway, especially as the original identity theft was because the dead camper bore some resemblance to our ID thief.
In answer to the first sentence, it's less common now than before, but I've attended funerals or viewings with open coffins. Your second sentence brought up something that I'd wondered about myself. The tip-off might be a facial scar that wasn't on the body, or the presence of all ten fingers on the imposter, whereas the friend knew that the original person had lost the tip of a finger back when they knew each other.
As something interesting and potentially useful for you: It's almost impossible to really positively identify the person in the coffin once they've been embalmed etc. They don't look like recently dead people. It really changes a person to be prepared for burial. When you look at a body you can see the person they used to be, you can see some of the same features and so forth there still and you say 'Ok that's them'. But even if you knew them well and had seen them recently you'd have a really hard time saying for sure picking your corpse out of a line up. Especially if someone is trying to deceive you and has put their doppleganger corpse in the right clothes and matched their hair color then I think most people would just buy into it. The real body doesn't look like they did when they alive, and if you haven't seen what the real body looks like post-embalming then you aren't likely to notice. It's only things like birth marks or tattoos or scars or similar that could really give them away because everything else has kinda changed a bit anyway.