Well, there IS a remarkable similarity to the shape of the the melody of “Lullaby of Birdland” and the first couple of measures of “That Old Feeling”. The two tunes also share a similar style/time history. Perhaps those are the things that you are reacting to, because Baker the trumpet player sounds nothing like Clifford Brown. Especially on the two tunes you link in which Brown plays with a mute (and very little).