How is an allergic response the same as a normal inflammation response

An allergy is a histamine reaction to a substance, usually a protein. This reaction can (usually) include inflammation. Inflammation is the reddening of an area of the body, usually accompanied by swelling, caused by increased blood flow to the area. Inflammation is part of the body's healing and defense mechanism.
An infection is a response to an infectious organism. This organism is causing a disease. It can be localized (such as a boil) or systemic. Most "infections" are caused by bacteria. Some infections can be accompanied by inflammation; If you have an allergy, you would take an anti-histamine, either OTC or by prescription. If you have an infection, and it is caused by a bacteria, you would take an antibiotic. If it is caused by a fungus, you would need an antifungal compound or drug. If it is caused by a virus, there is little medicine can do for you.