Because "especially" is an adverb, it should be used accordingly - which is to modify verbs, adjectives, and other adverbs. Furthermore, it is applicable when you either single out something because you like/hate/value/etc. something in a greater extent than others. Think of the word "particularly," since they have similar meanings. So the sentence, "I especially like reading," is grammatically correct, if you meant that you particularly like reading among other activities that you like (meaning, you singled out reading from the set of activities that you like because you like reading more). I hope this helps!

I'm not pretty sure, but with the sentence "I really like reading" you can end up the senstence. Instead, with the sentence "I especially like reading" you would have to specify what kind of readings you prefer reading. For instance, I especially like reading classical books or about science.

it is correct

Yes, 'I especially like reading' is a correct sentence. But no, it isn't just an alternative way of saying 'I really like reading'. As Jyn, says, you use 'especially' in the same way as 'particularly' - to emphasise the fact that you like one thing to a greater extent than other things that you have mentioned or implied. For example, you might say 'I like relaxing at home in the evenings. I like watching TV and listening to music, and I especially like reading.' This means that reading is your favourite activity of the three that you have mentioned.

I think it's Chinglish assuming that you are trying to express "我特别喜欢读书" in English. Although there is no any grammatical problem in this sentence, it does not feel appropriate. For instance, we say "I brought you a book." not "I brought a bood to you." The latter is grammatically correct, but still we don't say it like that. So, "I really like reading." will be good enough to express yourself. Hope it helps.