Traditionally this painting was wrongly identified as a portrait of Eleanor. She was in fact a bright bay with black legs and a white sock on her near hind. Her owner was Sir Thomas Charles Bunbury (1740-1821) whose colours were pink and white stripes, not red with white sleeves as seen here.
The gentleman in the long red coat bears some resemblance to one of the leading Newmarket trainers Richard Prince, although this scene is clearly not Newmarket. Prince's main owner, Colonel Henry Mellish, had colours of crimson sleeves and a white body (i.e. the reverse of what is shown here) with a crimson cap and white ribbon.