Firms that keep track are doing it as a customer service. Mo. Only the fair market value of the shares sold is reported. And there are some brokers who will tell you that their systems are not set up to allow for specific lot identification, which in this day and age is lame. But don't get nervous. Your records are most important.
Mutual funds are a bit different.
If you choose one method for selling shares of a particular fund, you must stick to that method. So if you've been selling your shares of Fund XYZ on a FIFO basis, you can't switch to specific-lot identification for your next sale. You can, however, use different selling methods for different funds. This method determines your cost basis by dividing the total cost of all the shares you own by the total number of shares. For all other methods, whether you're a stock or a fund investor, you don't have to file anything with the IRS if you use more than one method or even switch methods.