The power and ground pins will have continuity with each other when tested with a multimeter, but the SDA and SCL pins will not. This normal, and it due to these traces passing through the I2C level shifter component which does not behave like a simple point of continuity.
I have sometimes found that a breadboard can be the cause of certain mystery issues, they tend to wear down over time and can cause bad connections. It also appears as though you have soldered the pins such that the side going to the breadboard is shorter than normal, they may not be making proper contact. The plastic spacer should normally be on the side that will be mating- I know it doesn’t seem significant, but this could be a problem given sensitivity of I2C. I recommend trying an alternative connection method without a breadboard if possible.