Multiple professional organizations disagree (see below). For the societies that recommend age 45, they suggest that concerned women speak with their doctor regarding earlier screening.
Importantly, note that screening looks for signs of a disease in asymptomatic, normal-risk individuals. The presence of a family history or a lump/mass moves one from the category of screening into that of diagnosis.
Age 40: https://www.acr.org/About-Us/Media-Center/Press-Releases/201... - College of Radiology
https://www.acog.org/About-ACOG/News-Room/News-Releases/2017... - America College of Gynecology
Age 45: http://jamanetwork.com/journals/jama/fullarticle/2463262 - American Cancer Society
Age 50: https://www.uspreventiveservicestaskforce.org/Page/Document/... - United States Preventive Services Task Force