Elliott Gould (Streisand's husband at the time) was starring in the Broadway production of Drat! The Cat! in 1965 (it ran for 11 performances) with Lesley Ann Warren. 

Drat! The Cat! was a musical with book and lyrics by Ira Levin and music by Milton Schafer about a cat burglar (Ms. Warren) plundering New York society in the 1890's.

The big song from the show was “She Touched Me” and was sung by Elliott Gould. “I Like Him” was Lesley Ann Warren's big Second Act “eleven o'clock number” in the musical.

Streisand most likely recorded the two songs because her record company, Columbia Records, reportedly invested $50,000 in Drat! The Cat! (Record companies often invested in Broadway shows, hoping to recoup their investments by releasing and—hopefully—selling many copies of the shows' cast album.)

Although “He Touched Me” became a big hit for Streisand (and was included on her My Name is Barbra, Two album), “I Like Him” has never appeared on a Streisand Album, nor has it ever been released digitally.