Is the blood pressure in the lungs separate from the heart blood pressure?

Yes: Oxygen poor venous blood flows to the right side of the heart. It is then pumped into the lungs, and from there goes to the left side of the heart which pumps it into the arteries to the entire body. The commonly measured pressure is the arterial pressure from the lt side of the heart. The pressure in the lungs is from the rt side of the heart, is much lower, and takes an echocardiogram to measure.

Blood pressure: The heart pumps blood into two circulatory beds. One is the pulmonary or lung bed and the other is the systemic or 'rest of body' bed. Because the pulmonary bed has less flow resistance, the normal pressure in the lung bed is lower than that in the systemic bed. So yes the pressure in the lungs is separate from the pressure in the systemic circulation. Right heart pressure is lower than left heart.