Compound words made of two nouns often get a special ending at the end of the second word to show their relationship, that looks exactly like the third person singular possessive. "ingiliz" is being interpreted that way here, as a compounding noun.
Taksi (taxi) + şöför (driver) = Taksi şöförü "taxi driver"
Deniz (sea) + kenar (edge) = Deniz kenarı "sea shore"
Nope. the -i at the end of yemeği shows that it is a compound word construction with "İngiliz" (see my note above). It can receive any case ending after that.
(If it weren't a compound word, it would look the same in the accusative, i.e. "Ben bu yemeği yaptim." = I made this food. A lot of different suffixes in Turkish LOOK the same or create similar results.)
The third person singular possessive ending is -(s)I(n).
There's an (s) inserted between the word and the ending if the word ends in a vowel.
The (n) is used when you add more case endings to it.
Taksi şöfür-ü > Taksi şöfür-ün-ü gördum. (I saw the taxi driver)
Elma kurabiye-si > Elma kurabiye-sin-i yedim. (I ate the apple cookie)
Why can't Ingiliz yemegi be translated "English food" as well as "English meal"?