Not only.... but also

Is this sentence grammatically correct?

He doesn't only like football but also likes tennis.

and if it's wrong, why so?

Specifically, is there any problem with omitting the subject in the second clause?

Also, is there any problem with the verb form likes?

I would rephrase the sentence as

He likes not only football but also tennis

in the style of the first of the examples here. This rephrasing avoids repeating the verb after but also if it is the same verb that came before the not only (compare with the second example in the link, where there are two different verbs).

Note that here the not only... but also... construction is used to amplify the object of the sentence, which essentially is of the form "He likes THIS". That's why it is not necessary to repeat neither the verb "like" nor the subject "he" inside the object.

Analyzing : "He doesn't only like football but also likes tennis."

Normal usage is "not only X but also Y", but here "not" is getting merged with "does" , the usage looks odd.
More-over, "only like" refers to "like" , not "football", so using "tennis" later looks odd.

"He not only likes football but also excels in it."
Here, we have the normal usage of "not only X but also Y" & "likes" goes along with "excels", both in terms of football.

When you want to state that he likes football & tennis, then try this:
"He likes not only football but also tennis."
Here, "football" goes along with "tennis".

Reference :
The Cambridge Grammar Of The English Language :
https://archive.org/stream/TheCambridgeGrammarOfTheEnglishLanguage/The%20Cambridge%20grammar%20of%20the%20English%20language#page/n1327/mode/2up
Here, the common format of "not only X but also Y" is stated, with some possible variations.