I've only been trading stocks since 2005ish, but all the brokers I've dealt with have had ways to tell them which shares I was selling. And if I didn't pick them, they'd pick for me. Nowadays, they're required to track cost basis for regular shares. Even if you have shares where they won't report a cost basis, you're supposed to tell them which shares you're selling before the transaction settles.

Employment compensation related shares get weird, but they will at least track the purchase date or the date it entered their system anyway.