Does "Goodbye!" have any special meaning?
I wonder why when I say "Goodbye!" many people react as if I say "Goodbye forever!". I was sure there is not so much difference between "Goodbye!", "Bye!" and "See you!" and "Goodbye!" is just little more formal. But now I'm not so sure. Mostly I talk with non-native speakers but people from very different countries react this way. Can anybody tell me why could it be?

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As a native speaker, I see no issue with your usage of "Goodbye". Granted, in my personal experience, its more common for people to say simple "bye" and "see you" informally. But, there's no real issue with what you are saying, so don't feel like its out of place or anything. "Goodbye" is used a lot for sure.

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I agree with the other answers here -- there's no real difference between "goodbye" and "bye," but "goodbye" sounds much more formal, so most people only use it when they are saying goodbye for a long time, and it tends to sound more serious.  In my own region (southwestern US), I almost never hear anybody say "goodbye" at all.  Even at the end of formal or business phone calls, most people simply say "bye."

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Thank you very much, real_lampcap!

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This might be because we use simply "bye" more often that people react to "goodbye" as more formal and use it when they will be gone for longer periods of time. There is really no difference however because "bye" is just a shortened way to say "goodbye."