Is it possible to hold horizontal and vertical (similar to Ortho mode from Autocad) lines while using the measurement tool in QGIS?
I don't think that's possible unfortunately.
What you could do is create a scratch layer with line-type geometry and create a new feature with Advanced Digitizing turned on. This will allow you to draw horizontal and vertical lines and will display the distance as well. Not exactly what you're looking for but probably the closest to it.