I think I've understood your confusion. When you say "I don't have coffee, can I have tea ?" are you thinking that the first 'have' would be equivalent to "I don't drink/take coffee"? Perhaps my understanding was biased by having read the Japanese sentence first, but I think this would be quite a confusing thing to say even in English.
ない doesn't have that meaning. It denotes existence or possession. So the only reasonable interpretation of this sentence is "I don't have coffee. Will tea be okay?".