Learn Hardcore Korean: mogi apayo.

It can mean either, depending on context. In everyday health talk, 목이 아파요 is most often understood as “My throat is sore.” If you specifically mean muscle/neck pain, you can say:

For a stiff/sore neck: 목이 결려요, 목이 뻐근해요
For the back of the neck: 뒷목이 아파요
For the nape: 목덜미가 아파요

Why is there 이 after 목?

이 is the subject marker. It marks 목 (neck/throat) as the grammatical subject of the sentence. Use 이 after a noun ending in a consonant (like 목), and 가 after a noun ending in a vowel. So:

목이 아파요 = My throat/neck hurts.
If the noun ended in a vowel: 배가 아파요 (My stomach hurts).

What’s the difference between 이/가 and 은/는 here?

목이 아파요 simply states the fact that the throat/neck hurts (neutral/new information).
목은 아파요 contrasts or topicalizes: “As for my throat/neck, it hurts (but maybe other parts don’t).” Use 이/가 for neutral focus; 은/는 for contrast or topic-setting.

Can I drop the particle and say 목 아파요?