My mother in law does the same thing...to me. Every cold, cough or sniffle I have she swears are allergies. Fortunately, I don't suffer from seasonal allergies. I completely understand your irritation. It does get old to hear her diagnosis.
My kids do suffer from allergies. They do get cold like symptoms. Mostly a runny nose, sometimes a cough. NO fever. The pediatrician asked if they did the "allergy salute", which he demonstrated (basically the hand palm up swiping/ itching the nose). Both of my kids do this often. And my pediatrician says he can see a line on the inside of their nose where they are constantly rubbing it. He says it's a classic tale tale sign. My son also has bags under his eyes when it's a bad season. Doc says that too is a symptom. My kids don't sneeze or have itchy eyes for the most part.
Allergy testing was not recommended or needed in our case. I don't really need to know specifically what elements my kids are allergic to at this point. If their symptoms worsen or they are affected by more things, then the doctor will revisit the idea of allergy tests.
My son is affected worse, he takes an allergy medicine daily during spring and summer and "as needed" in the fall and winter. (We too are in Texas and you know we get random spring-like days often). My daughter isn't as sensitive and she takes medicine "as needed", which is daily during the spring for sure.
I'd simply ask their pediatrician if allergies are suspected the next time you go in for a well check or sick check. If for no other reason, to stop you mom from driving you crazy.