He is probably the Jacob aged 22 and 12 in these two census records (listed more recent first):

1860 census of Pendleton Co., Morgan P.O., enumerated 13 June, lists on p. 37 as family #243/244:
Joseph Montz, 57, b. Bonn, Germany, farmer; Lavina, 43, b. Penn; Jacob Davis, 22, farm worker, b. Ohio; Joseph, 18, b. Ky; Elizabeth, 16, b. Ky; Jacob Montz, 10, b. Ohio; George W., 5, b. Ky; Sarah M., 2, b. Ky. (Note that the household contains both a Jacob Davis and a Jacob Montz.)

This conjecture is based on the facts that his son Elijah’s death certificate says that he was born in Pendleton Co., Ky, and his son George Thomas’s death certificate says that Jacob was born in Cincinnati, Ohio, and that the 1880 and 1900 censuses say that Jacob and his father were born in Ohio, and his mother was born in Pennsylvania, and that the Joseph Muntz family later ended up in Harrison Co., Ky, where this Jacob Davis’s wife Catherine Jones was born, and that Jacob and Catherine named a daughter Lavina.

1850 census of Pendleton Co., Ky, district 1, enumerated 14 Aug, lists as family #127:
Joseph Munze, 41, shoemaker, b. Germany; Lavina, 35, b. Pa; Jacob, 12, b. Ky; Joseph, 7, b. Ky; Elizabeth, 5, b. Ky; Jackson, 2/12, b. Ky; Nicholas, 2/12, b. Ky.

Although at first this might appear that Jacob, Joseph and Elizabeth are children of Lavina and a previous husband named Davis, the birth and death records show that her maiden name was Davis:

1858 birth report for Pendleton Co., Ky, lists Sarah M. Montz, born on 16 Nov 1858, father Joseph Montz, mother’s maiden name Lavina Davis. (This would be the Sarah M. age 2 in the 1860 census.)

Death certificate for Joseph Muntz, age 77 years 5 months 7 days, born 1 Nov 1842 in Ky, died 7 Apr 1920 in Harrison Co, Ky, married, farmer, father Joseph Muntz born in Germany, mother’s maiden name Lavina Davis, born in Pennsylvania, cause of death apoplexy, informant Roy Muntz, Cynthiana, Ky.

This would be the person listed as Joseph Muntz age 7 in 1850 and Joseph Davis age 18 in 1860.

Death certificate for Jacob Muntz, age 71 years 5 months 9 days, born 5 Sep 1847 in Ohio, died 14 Feb 1919 in Cynthiana, Harrison Co., Ky, single, occupation shoe mender, father Joseph Muntz born in Germany, mother’s maiden name Vina Davis, born in Pennsylvania, cause of death cerebral hemorrhage, informant Joseph Muntz, Cynthiana, Ky, burial in Old Salem graveyard.

This would be the son listed as Jackson Muntz age 2 months in 1850 and Jacob Montz age 10 in 1860, not the one listed as Jacob Muntz age 12 in 1850 and Jacob Davis age 22 in 1860.

Since it is unlikely (although possible) that Davis is both Lavina’s maiden name and the name of an earlier husband, most likely either the Jacob born ca 1838 is a son of Lavina born out of wedlock before her marriage to Joseph Muntz, or he is really the son of a brother of hers, raised by her and Joseph Muntz. (Too bad the censuses did not start showing relationships until 1880.) Although his mother might be Lavina Davis, his father would not be Joseph Muntz, because the censuses are consistent that his father was born in Ohio, not Germany. Also Joseph and Lavina would be unlikely to name their son born in 1850 as Jacob, if the Jacob born in 1838 was also theirs.

1900 census of Fleming Co., Ky, Muses Mill precinct, enumerated 12 June, lists on p. 11A as family #177/178:
David, Jacob, 63, b. Mar 1837, Ohio Ohio Pennsylvania; Rebecca J., wife, 33, b. Feb 1867, Ky Ky Ky, married 20 years, 2 children, 1 living; Joseph A., son, 18, b. Nov 1881, Ky Ohio Ky.