Great question! The answer is that it depends. First let's discuss what "anxiety" means. Anxiety is when someone feels worry or fear for the future. This can take on many physical symptoms such as:
sweaty
heart racing
restlessness/ jittery
stomach tightness or nausea
feeling hot
lack of sleep
muscle tightness or tension
These are some of the most common experiences, but does not complete the list of body sensations that can be experienced. These can range from mild, moderate to severe depending on what is going on inside of the brain and body. Sometimes the fear is based on an event in the present or future, or sometimes anxiety is felt in a more of a general unease.
So when is it considered a mental illness? Anxiety is the brain and bodies natural way to keep us safe. It is meant to remember negative experiences to keep us on guard so we can keep ourselves safe from any "treat". This use to have a survival purpose to keep ourselves away from predators like bears, but now it is more about our value, our relationships and our school or work accomplishments.
Anxiety is considered a problem when it affects big aspects of someone's life. A "normal" anxiety would entail being worried about an important project or test, or event. However, if for example, someone experiences anxiety every time something is due or impending that can affect their quality of life on a more regular basis, then treatment would be advisable. To diagnose an anxiety disorder, it is required that the anxiety affects multiple aspects of their lives on a regular basis. The would include work, school, relationships etc.
You do NOT need a diagnosable mental illness to seek therapy services. Many people seek therapy for maintenance purposes, or to help with some techniques to feel better, even if the symptoms are at a "normal" level. Therapy can help you feel better in whatever your goals are!
With some skills and education, anxiety symptoms can be lessened or alleviated for a better overall quality of life.