Horseracing has trainers, but human sports have acquired coaches as well.
The term coaching with reference to human encouragement and advice appears to have origins in English traditional university "cramming" in the mid-19th century. (The name allegedly recalls the multi-tasking skills associated with controlling the team of a horse-drawn coach.) By the 1880s American college sports teams had, as well as managers, '\'coaches''. Some time in the 20th century, non-sporting coaches emerged: non-experts in the specific technical skills of their clients, but who nevertheless ventured to offer generalised motivational or inspirational advice.
When a coach works with an individual client, the first order of business is for the coach and client to work out a mutual understanding of the scope of work and document that understanding in a contract. Then the coach goes to work assessing the client's current performance and looks for ways to address the improvements needed.
Like individual coaching, the focus is on improving performance. In the case of a team, the coach observes the team's current functioning, assesses the team's strengths and weaknesses, and develops a plan for addressing the needed changes.