If you've had Hashimoto's for that long, have you also been on any replacement thyroid hormone medication? If so, what medication, dosage and for how long?
Have you had Free T4 and Free T3 tested to determine actual thyroid hormone levels? If so, what are their results, with reference ranges?
I've read that there's a connection between breast cancer and thyroid disease, but I'm not exactly sure what it is. I had an elderly aunt who was on Anastrozle for breast cancer and her TSH became elevated as well. I could never get them to test Free T4 and Free T3, though so I don't know what her actual thyroid function was. She was not on any thyroid medication. I don't know of a connection between Anastrozole and hypothyroidism, but that doesn't mean there isn't one.
Is TSH the only thyroid related test that was performed? Your doctor should have done Free T4 and Free T3 also. Those are the actual thyroid hormones and would give a better idea of actual thyroid function. TSH is a pituitary hormone and is not always indicative of thyroid function since it can be affected by a number of variables.
Insist on having Free T4 and Free T3 tested, then you can go from there.
Are you having symptoms of hypothyroidism? If so, what symptoms are you having? What treatment are you undergoing for breast cancer?