The post seems to have been lost in a server move but a few years ago I was looking at this article on wikipedia: Oldest football clubs - Wikipedia . I broke it down to Association Football only, and then filtered out clubs that are lying about their date of formation (e.g. Crystal Palace), and amateur teams, and as I recall we were among the 20 oldest in the world if we take 1874 as the start date. There are those who say that the foundation of the limited company counts as the starting date of a club: I don't necessarily buy this in all cases and particularly not in the cases of clubs like Morton and Kilmarnock, which, while incorporated in their current legal form until 1896 and 1906, respectively, but can still show clear articles of association from their foundation dates.
It does get muddy because then you have clubs like 1860 Munich saying they're from 1860, which is technically true because the current organization was founded then, but the football department didn't start until 1899. But what is clear is that Morton were founded as a football club in 1874 and enjoy an unbroken history to the present day, despite only becoming a limited company in 1896, three years after joining the Second Division.
Point is, 150 years as a professional club is a hell of an achievement, something very, very few association football clubs will manage anytime soon, and should be commemorated when it takes place in 2024.