The present tense of a verb has 3 different forms. Each form has its own special usage: I feel (simple) I am feeling (progressive or continuous) I do feel / I did feel (emphatic) When do/did is used with a verb, the simple present tense form of the verb is used (also called the bare infinitive).

Both sentences are incorrect grammatically. You would not use them in English. Sentence 1 should read "Why did I feel like that?" Sentence 2 should read "Why did I feel that way?" They can be taken to mean the same.