I thought tre was only used for adjectives so I looked it up in vortaro.net...turns out you're right, you can say "Mi tre ŝatas kafon".
I also looked at multe to see if I could learn clearly the difference between the two...but I couldn't, vortaro is still not super accessible at my level of esperanto.
What I brought home is that you can use multe to indicate quantity, duration (of an action or a thing); "Mi ne dormis multe hieraŭ". And tre in a more general way indicates superlatives, intensity. Also, you can say "tre multe" :D
This diferent is right, the verb "ŝati" the diferent between "multe" and "tre", but with "kuri" to run it becomes clear.
"Li multe kuri." He runs a lot, like everyday or for a long time, runs many times.
"Li tre kuri." He runs a lot, like his is fast, runs intensive.
With "ŝati" this diferences becomes the same, a big amount of likeness or a strong likeness is the same thing.
There are some good answers already posted. I would explain it as follows.
Tre has to do with intensity and multe has to do with quantity. Whether you use tre or multe is a question of semantics and not grammar. In many cases - such as this one - you can use either.
- My appreciation of coffee is intense.
My appreciation of coffee is gigantic.
I really like coffee.
I like coffee a lot.
Mi tre ŝatas kafon.
- Mi multe ŝatas kafon.
There are some words where you must use one or the other (for semantic reasons.)
- Mi estas tre laca - I am really tired. (Not "I am tired in a large amount")
- Mi multe promenis - I walked a lot (Not "I very walked.")