Faulty English translation, "salad" is no person! Use "of"-phrase here. And, to my mind, the phrase "the tomato of the salad" doesn't make much sense in English anyhow. An English speaker would probably use the phrase "tomato in the salad" instead. (Of course the matter can be looked at more detailed-wise. Animacy, possessive relation, and more. But this does not change the argument concerning "the salad's tomato")
But we are not trying to say ...." the tomato salad"...we are referring to the tomato which the salad contains. And btw in both AE and BE it is "tomato" the plural takes the "e"...."tomatoes" Here it is in
the Cambridge Dictionary...pretty British right>
Check out the British English here
this from Oxford https://www.lexico.com/definition/tomato
If we wanted to say "the tomato salad" we would definitely say "the tomato salad" .