My gt.grandfather changed his name and I think it's quite possible your ancestor did too. My g/g had his two other names Franz (Frank) and Jacob to choose from but his guardian chose to amend young Henning's name to Herman.
I think we have to look at the area he migrated to; the pronunciation of words in the area, at what age and if he could read & write. If a boy, was he teased with a nickname such as a 'cob' (a loaf-er), a 'cob' (a type of horse). Did he live at one time in a mixed French/ Dutch / British quarter where the first syllable Yac/Jac became a cause of teasing, e.g. a 'yak which was a beast of burden? Do you know Jacob's other names - was one of them Johann? This would translate to 'Jean' pronounced with a soft 'j' in a French quarter.
Another simple reason could concern his workplace; maybe a longtime employee had the same names as your ancestor and it was decided Jacob's forename should be changed. Someone has already suggested Jacob/Jacque could have been interpreted as 'Jean'.
Yes, Jack is the diminutive of John but Jacob is the Hebrew version of the Greek name Iakobus, which became the Latin Jacobus - this latter became Jacques in France which in turn was the English name James (which I find hard to believe too!).
The best thing is to keep an open mind when researching all options.
Best wishes,
Rena
Is it likely that the name Jacob for person born in Germany be
changed to Jean after arrival in America?
Max