Prior to 1968, this fact didn't seem to bother anyone and things were running along pretty smoothly in the birthday celebration department -- February 22nd was observed as a federal public holiday to honor the birthday of George Washington and February 12th was observed as a public holiday (in most states) to honor the birthday of Abraham Lincoln.
Then things changed. In 1968, the 90th Congress was determined to create a uniform system of federal Monday holidays, so they voted to shift three existing holidays (including Washington's Birthday) to Mondays. The law took effect in 1971. As a result, Washington's Birthday holiday was changed from its fixed February 22 date to the third Monday in February. This change was not without controversy. There was some concern that Washington's identity would be lost (since the third Monday in February would never fall on his birth date of February 22nd). There was also an attempt to rename the public holiday "Presidents' Day", but this stalled in committee. "It was the collective judgment of the Committee on the Judiciary," stated Mr. William Moore McCulloch (R-Ohio) "that this [naming the day "President's Day"] would be unwise. Certainly, not all Presidents are held in the same high esteem as the Father of our Country. There are many who are not inclined to pay their respects to certain Presidents. Moreover, it is probable that the members of one political party would not relish honoring a President from the other political party whether he was in office, no matter how outstanding history may find his leadership."
The single holiday observance meant that the traditional 10-day separation between Washington's Birthday (February 22) and Lincoln's Birthday (February 12) had essentially been eliminated. However, while Congress had created a uniform federal holiday law, there was not a uniform holiday title agreement among the individual states. Even though most states with individual holidays honoring Washington and Lincoln shifted their state recognition date of Washington's Birthday to correspond to the third Monday in February, some states, including California, Idaho, Tennessee, Texas and others, chose not to retain the federal holiday title and renamed their state holiday "President's Day."
From that point forward, the growing use of the term Presidents' Day was largely a marketing phenomenon, as advertisers sought a catchall phrase to capitalize on the opportunity for three-day or weeklong sales. Gradually, the phrase "Presidents' Day" took hold and today has become part of the everyday vernacular. Interestingly, in 1999, bills were introduced in both the U.S. House (HR-1363) and Senate (S-978) to specify that the legal public holiday once referred to as Washington's Birthday be "officially" called by that name once again. Both bills died in committees.
Check out this blog post from the National Archives that references their article on Washington's birthday.