He appears to be a distant relative, yes, but by my count, five generations removed from any direct connection to the banking business.
Starting with his entry [1] on thepeerage.com, a UK genealogical site for the upper classes, you can backtrack to find that his great-great grandfather was a Charles James Hoare [2], who appears, if this Wikipedia entry is the same person [3], to have been a a Church of England clergyman and himself the son of a partner in the Hoare private bank. So, C.A.R. Hoare's great-great-great grandfather was part of the bank. The four generations in between seem to have been various kinds of upper-middle class government or quasi-government employees though: clergyman, clergyman, military officer, and colonial civil servant.
Incidentally, I got the pointer to thepeerage.com from his biography in the book Reflections on the Work of C.A.R. Hoare, which calls his family background "somewhat upper-class" and references the site in a footnote.
> He developed the sorting algorithm quicksort in 1959/1960. He also developed Hoare logic for verifying program correctness, and the formal language communicating sequential processes (CSP) to specify the interactions of concurrent processes (including the dining philosophers problem) and the inspiration for the occam programming language.
