Are bug bites the same thing as allergies?

No. Bug bites are not the same as allergy. Most folks who have a bug bite (mosquito) will have local swelling and itchiness that resolves relatively quickly.
They Can Be... Mosquito bites, although not life threatening, can be ige-mediated (to the salivary enzyme in the female mosquito) and cause large local reactions which persist for several days with symptoms of pain, itching, swelling and redness. The best treatment is prevention of the bites using insect repellants. I find that eucalyptus oil works very well in preventing mosquito bites.
No. Bug bites often cause hives at the site of the bite, sometimes can be large, due to components of the insect bite. Allergies to bug bites or other elements are mediated by elements of the immune system, such as the ige, the antibody that triggers the immediate allergic reaction resulting in itching and generalized hives, and sometimes leads to anaphylaxis.