From now on, therefore, we regard no one according to the flesh. Even though we once regarded Christ according to the flesh, we regard him thus no longer. (2Cor. 5.16)
So, “to know someone according to the flesh,” really means to know them from the perspective of my natural thinking – which would be to view them from the perspective of myself as the center of my universe. It would also mean that I know them based on their natural traits, personality, and fleshly characteristics. In other words, Paul is talking about relationships that exist fully in the NATURAL realm, rather than ones that are based on the new creation in Christ.
But Paul tells us that all of that is wrong – it is a wrong way of looking at things because of Jesus Christ. We are not the center of the universe. Christ is. And the REAL person we are, and the REAL person that someone else is, is NOT the natural man that we see, or think we see. No. The REAL person is found only in Christ Jesus.