Yes, anxiety disorder is a mental illness. Mental illness is also known as mental disorder or psychiatric disorder. Mental illness is a brain disorder, and are diagnosed and treated by psychiatrists. Since there are no current biological markers to confirm if someone has a mental illness, these disorders are diagnosed by history and clinical presentation. Mental illness such as anxiety disorder have detrimental impact to your life, as it disrupts your work, schooling, relationships, and ability to care for yourself. Even though anxiety disorder is a mental illness, it is a milder form of mental illness, whereas schizophrenia and bipolar disorder are more severe mental illnesses. Depression is also a mental illness, but similar to anxiety disorder, it is a milder form of mental illness. So for mild mental illnesses such as anxiety and depression, you may start treatment with psychotherapy, and if that does not work alone, then prescription medication may be added. More severe forms of mental illness such as schizophrenia and bipolar disorder need prescription medication as the primary treatment, then psychotherapy can be added once the severe mental illness is stabilized.
So mental illness such as anxiety disorder is different from mental distress and mental health problems. Everyone experiences mental distress, as it is part of your everyday life. We all have disappointments, failures, accidents, which can occur on a daily basis. As an example, mental distress could be caused by your boss yelling at you at work; it could be you lost your keys; maybe you are late for an important appointment. So mental distress is normal, everyday disappoints and mishaps. You can usually cope with mental distress and work through it. Anxiety disorder is not mental distress (everyday mishaps and disappointments), as anxiety disorder is mental illness (a brain disorder).
Mental health problems are more severe than mental distress, but less severe than mental illness. Mental health problems come about when you suffer from a major stressor in your life. It could be you had a death in the family. Maybe you got fired from your job. You might be going through a separation or a divorce, or you had a breakup with your boyfriend or girlfriend. These major life stressors are experienced by many people, and when they occur, they can be very distressing. People with mental health problems brought about by major life stressors often need to see a counselor to build coping skills and to talk through their stressors. Mental health problems can turn into mental illnesses such as anxiety disorder and depression if the major life stressors are not adequately dealt with.