But there are a couple of problems with celebrating Smith's birthday on June 5th. First we don't actually know Smith's date of birth, no matter what his gravestone says. June 5 is the date of his baptism.
Ian Simpson Ross writes in his book "The Life of Adam Smith",
The Fife seaport of Kirkcaldy, ten miles across the Firth of Forth from Edinburgh, was the scene of Adam Smith's baptism on 5 June 1723, in the Old Parish Kirk of St Brisse (Bryce). Possibly this was his birth-date, though there is no annotation on the 'Register of Baptismes in the Kirk of Kirkcalsie' (Bonar, 1932: 208), stating 'born this day', as there is in the case of Smith's great friend David Hume (Mossner, 1980: 6). It is reported, however, that as an infant Smith was 'infirm and sickly' (Stewart I.2), and understandable anxieties of the time about infant mortality and salvation may have hastened baptism (Flinn, 1977: 284). (Emphasis added)while Gavin Kennedy writes in his "Adam Smith: A Moral Philosopher and His Political Economy",
His son, the world-famous Adam Smith, was baptised on 5 June (his birth date is unknown; old calendar; Bonar,  1966, 208).It is possible that Smith was baptised on the day of his birth, but it is also possible he was not.
There is a second problem with celebrating June 5th. The date 5 June is the date from the Julian calendar. Scotland continued to use the Julian Calendar until 1752. It was only then that the Gregorian calendar was commonly used. Using the Gregorian calendar, as we do today, Smith's date of birth, assuming June 5th to be correct, would be June 16.