This trend started with what's now called "Web 2.0" around 2000. Before that ("Web 1.0") websites were largely used only for information dissemination: here's who we are, what we do, how to contact us. As interactive technologies became more powerful and companies started hiring Web development teams, we started to see personal "portals" that allowed users to see or do something specific to their account or relationship with the company running the site. It was novel enough at the time that developers differentiated these features with the "My" prefix.
Nowadays we all expect the websites for our bank, utility company, phone provider etc. to have individual account management features so it just sounds lame.