His, her, and its are possessive pronouns, which are single words.
He's, she's, and it's are contractions of short phrases ending in "is"
Elephant's (as in the elephant's foot") is a possessive adjective, and may have originated as a contraction.
The oldest English grammar book in my collection is John Wallis's Grammar of the English Language (1653), [Tractatus De Loquela & Grammatica Linguae Anglicanaea], new edition with translation [from Latin] and summary by J A Kemp, Longman 1972. See pp 305 - 308.
On p 308, "Some people believe that the s is added instead of the word his ... and that an apostrophe should always be inserted or at any rate understood" (translation).