I know skills is a plural, but I am confused because I think English reading is a singular word not plural. Why English native speakers say English reading skills, listening skills, and writing skills? Instead of reading skill, etc.
Reading is not being considered as singular or plural in this case. It's being used as an attributive noun (or noun adjunct), which functions adjectivally, in the formation of the compound noun.
The only word that is playing a role in the singularity or plurality of the phrase is skill:
I have a skill.
I have skills.
(With the singular, the indefinite pronoun needs to be used. With the plural, words like many, some, several, or count words—three, eight, and so on, can also be used in front of skills, but it's optional.)
It doesn't matter what descriptive noun (or actual adjective) you put immediately in front of skill, the same thing will apply:
I have a useful skill.
I have two useless skills.
Also, attributive nouns normally (although not always) take a singular form.
✔ I ate two ham sandwiches.
✘ I ate two hams sandwiches.
So, in the example in the question, the choice in terms of grammar comes down to:
✔ You have a good English reading skill.
✔ You have good English reading skills.
✘ You have good English reading skill.
Why we have chosen to always use the plural version is a matter of speculation. I suspect that in school, reading skills is a term that encompasses many different areas: vocabulary, speed, comprehension, grammar, punctuation, and so on. Since there are many different areas, the plural form makes more sense.
It's possible to use the singular version, but doing so would often mean (idiomatically) rephrasing the sentence:
You are skilled in English reading.
Note how that changes the grammar. English reading, or more specifically reading, becomes an actual noun, rather than just an attributive noun.
It's also possible to make it plural:
Nancy travels across the country, giving English readings to an enthralled audience every night.
In this sentence, readings (with English remaining in its role of an attributive noun) becomes plural—because her nightly presentations are being counted.