There is no pre-existing function, to my knowledge.
Meaning you'll have to implement that yourself using pyqgis or maybe even creative use of the field calculator.
Note that the Arcgis tool uses a more elaborate approach as it is meant for complicated polygons. In your case, where each polygon seems to be just a buffered line, you could simplify this a lot. The logic could be:
For each poly
- Get the bounding box
- Get the long side of the bbox
- Get the angle of it
Bonus: You could easily calculate the certainty of the angle by calculating the ratio of long / short side of the bbox.
If you need help with implementation, follow the advice in PolyGeo's comment and ask as precise questions as possible when you're stuck.
There actually is a tool in QGIS for this!
Apply the Oriented minimum bounding box processing tool. It will create exactly what it says it will, and bonus: It already gives you the angle as an attribute field on the resulting geometries!
Remember this is not the same approach that the ArcGIS tool uses that you reference, as explained above!