They are the same persons. Many people "Americanized" their names after they emigrated to the USA.
Often because their original names were difficult to pronounce in English. The way you pronounce the "sch" in Bisschop has a tone that is not found outside The Netherlands and will be changed to Bishop in the USA.
Klaas will become Charles,
Jan will become John and as for Grietje;
the name Grietje originally derives from Margareth(a), short Maggie, and is also often "translated" as Grace.
I have often seen different names for the same persons in the USA as time progresses, this is not unusual.
And the date of birth of Charles on 20 November 1866 you mentioned, Klaas was born on 18 November 1866. Being off a couple of days is something I have seen time and time again.
So yes, John and Grace/Maggie with son Charles are the same as Jan and Grietje with son Klaas. No doubt about that!