Fitness trackers use a technology called Photoplethysmogram to record heart rate which basically flashes a green light on your skin and it reads how much light reflects back. The volume of blood under the skin goes up and down with each pulse, so the device can read these changes and determine heart rate.
Accuracy of these devices fluctuates depending on activity and device.
We saw no statistically significant difference in accuracy across skin tones, but we saw significant differences between devices, and between activity types, notably, that absolute error during activity was, on average, 30% higher than during rest. Our conclusions indicate that different wearables are all reasonably accurate at resting and prolonged elevated heart rate, but that differences exist between devices in responding to changes in activity.
Another study looked at different wrist devices (the PusleOn and Empatica E4) and it noted:
The accuracy and reliability of the devices were decreased during household work due to the excess hand movements.... The percentage of correctly detected heartbeats was 89% for PO (PulseOn) and 68% for E4 (Empatica E4) during sitting but 76% for PO and only 9% for E4 during household work.
The point is that these devices are subject to a lot of variability and error. Some devices seem to be quite good at it, but others not so much. They should be used more as guides rather than absolute authorities on what your HR is. Chest straps and arm band style heart rate monitors use the same technology, but they are better designed for monitoring so they tend to be more accurate.
I can't speak to why your wrist monitor records jogging better than climbing a steep hill, but it seems apparent that the difference in movement is enough that it struggles to accurately read climbing hills.