Fühlen can either take
- transitive form with an accusative object - "Ich fühle die Sonne auf meiner Haut"
- be reflexive with an adverb - This is how you feel - "Ich fühle mich heute schlecht"
Only in colloquial German you can use the reflexive form without an adverb or an adverbial construct.
Er fühlte sich heute nicht und ist heim gegangen
That one is a shortened form of "Er fühlte sich heute nicht gut" - In some regions you're apparently allowed to leave off the "gut". This one I have heard in the Cologne and Ruhrgebiet area, but never in the South.
Ich fühle da wie du
(The second one aligns with your last example - I consider that colloquial. It literally translates to "I feel like you" which doesn't make a lot of sense to me because it makes me want to ask "what?")