In the UK certainly, for a given request there'll be a clinical question to be answered (e.g. ultrasound liver for ? gallstones, CT brain for ?tumour), and the interpreting radiologist will have answering that question on their mind when reviewing the test. Having said that they will (or should) review all the data presented in the imaging, so e.g. an ultrasound of the liver is in fact an ultrasound abdomen and will contain images of the kidneys spleen bladder etc. An MRI of the spine for ? disc problems will yield info on the bones, muscles and nerves in that region, as well as info on the discs themselves.
Many a random cancer has indeed been picked up during imaging for an unrelated condition. And as pointed out, different radiologists/ doctors will interpret image findings a bit differently.
Cheers.