Anaphylaxis is a fancy word for a severe allergic reaction. In a severe reaction you get breathlessness, wheezing, choking, or you may collapse and lose consciousness. You usually don’t see all these things at the same time, although it’s possible. Severe swelling of lips, eyes, tongue etc can look terrible but is rarely life threatening so we don’t consider it anaphylaxis without the other things.
These kinds of reaction are unusual, but can be life threatening. We have particular ways of treating these reactions, including adrenaline pens (eg EpiPen), but of course it’s better to prevent such reactions ever happening if possible.