Insect bites can sometimes be dangerous to your health for many different reasons. True allergies to insect bites (other than bee stings) are not that common. Thus even though you are allergic to bees, I think that it is a low likelihood that you are allergic to other insect bites such as spider bites. There is no way to tell unless you react badly to a bite sometime down the road.
Other insect bites can cause other problems such as infections. Often a bite from a spider or other bug can cause a small infection under the skin that may need to be treated with antibiotics. In addition, some insects such as ticks can carry dangerous infections with them such as the bacteria that causes Lime disease.
Since you are allergic to bees, you should always carry a syringe with epinephrine (called and epipen) to inject yourself with if you get stung and have trouble breathing. I also suggest that you schedule an appointment with your primary care physician
. The two of you can discuss your allergy to bees and what the real dangers are. You can also discuss what other insect bites you should be worried about in your area of the country.