If you look at the facts in your post, you answered your own question.
1) I suspect like many people with HA, you try to "convince" the people around you that you're not imagining your worst fears. Your mom has had it with being told you have terrible illnesses, so she's decided to tell you you're right. It's not a great response for someone with HA, but it's a place that our loved ones frequently get to. The important thing is: She's not telling you you have bc because you have bc... she's telling you that because she's fed up.
2) Yes, fibroadenomas are common in young women. Much more common than bc.
3) If you had a palpable bc tumor that had been untreated for 3 years, it would either be huge or you would be dead. It hasn't grown in 3 years - if it looks like a fibroadenoma and it doesn't grow like a fibroadenoma, it's a fibroadenoma.
So you have two choices: You can get your regular ultrasound, which will show no growth, or you can trust your doctor that's it's a fibroadenoma. If this doc is a gynecologist (which you should get sometime soon if you don't have one already), she can also switch to a clinical exam once or twice a year to assess if it changes (which it won't).