Battery voltage drops when connected to PCB
Is that a fundamental rule to PCB designing that i am missing?


NO.

Simple as that.

There are, however, differences. The breadbord is pretty bad, from an electrical engineering perspective. It adds stray inductance, capacitance and resistance everywhere. This usually makes your circuit worse. In this case, maybe by pure luck, the exact same design on the PCB oscillates because it now lacks the stray capacitance and inductance, but I wouldn't bet on it.

Oscillation in a circuit can make things draw much more power than expected, but you won't be able to verify this easily without an oscilloscope¹. Without an oscilloscope you have limited means to debug this issue.

If you power this with a battery, it sounds to me that you don't have a well-equipped lab, but you do have a multimeter. I don't know what you mean by "verified" that the PCB and breadboard are the same, but you should check for shorts with a multimeter, not just look at the PCB.

1. pun not guaranteed