Sounds like some kind of infection? Maybe someone on here can help you.
Go to a different vet and get a second opinion. I have never heard of a vet who would just say "I don't know what that is, you need to put him to sleep." Anyone who would make a diagnosis like that doesn't need to be in practice. As much as you love your cat, you owe it to him to take him to someone who is at least going to make the effort to figure out what is going on with him. It could be something simple.
If I had to guess at what it could be, it sounds to me like your kitty is producing an overload of porphyrins. Porphyrins are secreted at times of stress as reddish brown goopy "tears" and around the nose, and it sounds to me like this might be what is going on with your kitty. Only a visit to the vet will tell for sure, but the one you went to should definitely be crossed off the list of competent veterinarians in your area.
Please let us know what happens.
Ghilly
I totally back Ghilly!! This makes me soooo dang mad when vets tell ya to put an animal to sleep because the animal is old. They think the problem is not worth investigating. The vets that are that lazy and ignorant should be the ones to be put to sleep. LOL
Get a second opinion. I would tell the vet at the very beginning of the appointment that you don't want to waste your money if they're just gonna tell you to put your cat to sleep without investigating. I'm not kidding tell them this before you put out the money. You might find a problem that's fixable and you could get a few more years with your baby.
I did have a similar thing happen with one of my cats-he had the brown junk around his nose and eyes, too. I would just wipe it everyday and realized that it was old caked up soft food that he didn't wash off. My cat was eating and drinking. So, that could be part of your cats problem that he's just not washing his face. Since he's not eating, this is worth looking into.
The problems worth looking into with older kitties are-kidney disease (urine and blood work), thyroid disease, diabetes. Or, it might be an infection.
I would request an x-ray, full blood work that looks at full blood count w/ liver and kidney panel, thyroid (blood test) and diabetes (blood). Also, get his teeth looked at. This is important with older kitties, if one or more teeth are bad the cat won't eat.
With these tests, I am pretty sure that you'll get some, if not all, your answers. I am sure that the brown gel gook is old food. I am glad that you did not listen to that idiot of a vet that told you to just put the cat down. He was an insensitive jerk!!
Please keep us posted and we will keep you and kitty in our prayers.