Mounts should be correct. Without further context, either could be correct.
He mounts his fathers horse, then he issues commands to the soldiers. You do not know from this one sentence if he is moving merely mounting or riding.
It truly bothers me how Duolingo uses such low frequency sentences in their method.
How about: "The flag of my country/My country's flag', "The name of my child/My child's name." There are more examples. However, you are right it sounds very strange but only to a native speaker who does not yet understand the structure of the second language. As you gain fluency in the second language they should make sense. I disagree with the statement of 'thinking in another language.' A horse rider in Brazil will not have a different experience as a horse rider in Mongolia. They both ride on top of the horse however differently they may express it.
I did not mean to imply that thinking in another language automatically brings to mind different experiences. What I mean is that when you think in English, for example, and then translate th thought to express it in a second language, you tend to carry the English grammar and structure into your spoken expression. I learned Spanish as a toddler and started learning English at age 6 in New York City public schools. I attended high school and college in Puerto Rico, and later did another 8 or 9 years of both undergraduate and graduate work in New York. I am fully bilingual. My brother-in-law spent all of his life in Puerto Rico and was transferred to Connecticut 5 years ago to run a division for his firm. Although, he is fluent in English, there is a significant difference between his English speech and mine. Certainly, he has a strong accent but he also has difficulty with sentence structure. I see the same issue in people raised in New York who speak Spanish using English structure.