According to a review of Spurzheim, Philosophical Catechism of the Natural laws of Man in the American Baptist Magazine, (April 1833), the analytical system of Dr. Johann Spurzheim categorizes the "affective faculties" into two "genera":
The propensities. Desire of life, desire of meat and drink, sense of amativeness, sense of parental love, sense of attachment, sense of habitation, sense of courage, sense of secrecy, sense of acquiring, sense of constructing.
The sentiments. Sense of cautiousness, sense of approbation, sense of self-esteem, sense of benevolence, sense of reverence, sense of firmness, sense of conscientiousness, sense of hope, sense of marvellousness, sense of the ideal or perfect, sense of mirth or humor, and sense of imitation.
Thus, for Spurzheim, describing humor as a "sense" amounts to describing it as a sentiment, coequal with such other senses as self-esteem, benevolence, reverence, and hope. But Spurzheim also uses sense to describe some of the entries on his list of "propensities"; and in everyday English, the modes of perception are called senses as well.
Given that English has for centuries used "sense of" to mean any of a number of ideas—including "feeling of," "consciousness of," "awareness of," and "notion of"—it is hardly surprising that the idea or recognition or appreciation of humor should also be described as a "sense of humor." In a thoroughly unscientific series of Google Books searches (the results of which are presented below), the first occurrence of "sense of humour" was from a book published in 1753—by no means early in the continuum of "sense of X" entries that I checked, and much later than such other mental senses as sorrow, love, piety, shame, gratitude, grief, and joy.
A Chronology of Senses in Google Books Search Results
I ran a Google Books search for the years 1550 through 1900 for one hundred "sense of" phrases, each of which yielded matches from not later than 1880. The phrases I searched for were as follows, in chronological order by first Google Books match:
1584: "sense of feeling"
1590: "sense of sorrow"
1591: "sense of love"
1598 "sense of hearing"
1603: "sense of pain"
1604: "sense of smelling"
1619: "sense of piety"
1623: "sense of shame"
1626: "sense of wrong"
1628: "sense of plesure"
1631: "sense of gratitude"; "sense of sin"
1633: "sense of grief"
1634: "sense of discipline"; "sense of faith"; "sense of frailty"; "sense of comfort"
1640: "sense of pity"
1641: "sense of security"; "sense of scruple"
1642: "sense of seeing"; "sense of humanity"
1643: "sense of religion"
1646: "sense of want"
1648: "sense of joy"
1649: "sense of tasting"; "sense of touching"
1653: "sense of duty"; "sense of mercy"
1654: "sense of conscience"
1658: "sense of compassion"
1659: "sense of honour"; "sense of peace"
1660: "sense of sacrifices"
1674: "sense of goodness"
1692: "sense of danger"
1694 (from a 1657 journal entry): "sense of truth"
1699: "sense of propriety"
1700: "sense of honesty"
1702: "sense of virtue"
1705: "sense of modesty"; "sense of merit"
1710: "sense of faith"
1714: "sense of vice"; "sense of right and wrong""
1719: "sense of the sublime"
1726: "sense of beauty"; "sense of obligation"; "sense of fear"; "sense of mortality"
1728: "sense of happiness"
1729: "sense of law"
1730: "sense of justice"
1740: "sense of devotion"; "sense of perfection"
1748: "sense of benevolence"
1749: "sense of approbation"
1751: "sense of right"; "sense of imitation"
1753: "sense of humour"
1757: "sense of impropriety"; "sense of awe"
1769: "sense of proportion"
1771: "sense of salvation"
1773: "sense of irony"
1785: "sense of self-esteem"
1787: "sense of godliness"
1790: "sense of pride"
1792: "sense of seriousness"; "sense of understanding"
1795: "sense of reverence"
1797: "sense of responsibility"; "sense of sympathy"; "sense of perdition"
1800: "sense of surprise"; "sense of righteouness"
1802: "sense of respect"
1806: "sense of admiration"
1815: "sense of fair play"
1819: "sense of attachment"
1821: "sense of wonder"
1823: "sense of serenity"
1825: "sense of conscientiousness"
1826: "sense of abandonment"
1829: "sense of mockery"
1831: "sense of hope"
1833: "sense of caution"
1834: "sense of outrage"
1835: "sense of trust"; "sense of longing"
1839: "sense of despair"
1840: "sense of appreciation"
1841: "sense of approval"
1842: "sense of absurdity"
1849: "sense of caution"
1851: "sense of genius"
1853: "sense of pathos"
1873: "sense of fun"
1880: "sense of disbelief"; "sense of play"
Not surprisingly, the earliest instances tend to involve relatively basic and/or devotional senses; and not surprisingly (to me), negative senses such as sorrow and pain tend to precede their positive counterparts such as joy and pleasure, sometimes by decades.