Learn Hardcore Turkish: O, aynı soruyu sorup duruyor; nazikçe yanıtladım. - She keeps asking the same question; I answered politely.

It’s the pattern V-ıp durmak, which means “to keep V‑ing; to go on V‑ing,” often with a hint of persistence or mild annoyance. Here, sorup duruyor ≈ “(he/she) keeps asking (and won’t stop).” The verb durmak is an auxiliary here; it doesn’t mean “to stand” literally.

Why is it sorup, not sormaya?

sorup duruyor uses the converb -ıp/-ip/-up/-üp to chain the action to durmak: “ask-and keep (at it).”
sormaya devam ediyor uses a verbal noun plus devam etmek: “continues to ask,” which is more neutral. Don’t mix them: “sormaya duruyor” and “sorup devam ediyor” are ungrammatical for this meaning.

Why is it aynı soruyu with accusative case?

Because the direct object is specific/definite: “the same question.” In Turkish, definite direct objects take the accusative -(y)ı/-(y)i/-(y)u/-(y)ü. So we say aynı soruyu. Without the accusative here, it sounds wrong to most ears.

Where does the extra y in soru-yu come from?

It’s the buffer letter y (kaynaştırma harfi). Since soru ends in a vowel and the accusative suffix also starts with a vowel, y is inserted: soru + (y)u → soruyu. Vowel harmony selects -u because the last vowel of sor- is back and rounded.