There is another Encyclopaedia which says 1873.
If you asked the author of that encyclopedia as to where they got the info from they'd say "a book".
Likewise if you asked the people who decided that 1973/74 was the Centenary year.
If you challenged them to come up with a shred of real contemporary evidence they'd be lost because it was only ever received wisdom.
The error seems to have cropped up around the mid-1930s. Even as late as 1932 the Carnival booklet (which includes a couple of historical of articles) says that the origins of the club are wrapped in mystery and it seems to have just grown up. In other words they didn't have a clue about the very early years. Jack Smith seems to have had a bit to do with putting together that booklet but he didn't join the Chemics til 1880.
Presumably because the prestige of the club kept growing in the 1930s the demand must have been there to come up with a definitive foundation date (what with articles for Wembley programmes and other big games) so it is likely that 1873 was guessed at.
One of the founders mentioned in the 1875 article, Nat Farrant, didn't die until 1936, indeed he was at the Wembley final of 1934 supporting the club he helped to found despite having left the town to become a succesful businessman in Yorkshire some half a century earlier. It's likely that Nat or some other old-timer was asked to come up with the formation date and, after the passing of 60 years or so, misremembered.
Once it appeared in print the mistake kept being repeated in ignorance of the archive evidence for the 1875 . Having said that one programme (maybe Wembley 1950, can't remember off hand) does mention 1875 but that might have been a typo!