Actually there is no irony here. We don’t know Smith’s date of birth. June 5 is most likely the date of his baptism.to which Don Boudreaux kindly replied
Not quite. Smith was born a sickly infant and, many historians believe, was likely baptized immediately rather than, as was the custom for more-healthy newborn babies, a few weeks later.Just how "sickly" would have Smith had to have been on the day of his birth for a baptism to be arranged the very same day? We know Smith was unwell as a child but how early this was manifested itself is, without further documentary support, speculative. And we have no evidence that he was baptised on the day of his birth. Ian Simpson Ross writes in his book "The Life of Adam Smith",
Here’s Nicholas Phillipson, on pages 16-17, of his 2010 biography of Smith entitled Adam Smith: “Smith was by all accounts a sickly child, and it may be that the slight confusion about his birth day arises from his having been baptized on the day of his birth – a common enough practice in the case of infants not expected to survive.”
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).So it is possible that Smith was baptised on the day of his birth but it is equally possible he was not. On the evidence I would go with Kennedy and say we know his date of baptism but his date of birth is unknown.
There is a second problem with Boudreaux's claim, 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. But Keynes's birth date is, of course, recorded using the Gregorian calendar and Smith's birth date under this calendar would be 16 June. So again the birthdays would not coincide.
Update: At his blog, Adam Smith's Lost Legacy, Gavin Kennedy author of "Adam Smith's Lost Legacy " and "Adam Smith: A Moral Philosopher and His Political Economy" writes,
The 5 June, 1723, is the date on his baptism certificate; his birth date is not known for certain. Also, 5 June is under the old calendar; the modern calendar (from 1752) places it on 16 June, as pedants keep informing me. However, I think it best to stick to the convention of the date on the only known certificate, because retrospectively changing every date on every document before the date was changed by a modern calculation would make historical work prior to 1752 a hopeless mess, as anybody working with 18th century documents will tell you.