It is more common in British English. To use "sore eyes" for certain situations.
"if you get grit in your eyes, they would not be strained" you would say "my eyes are sore" or "I have got sore eyes".
if you have been using your eyes for long periods or maybe trying to read without spectacles. you would say "I have strained my eyes" "My eyes are tired and strained"
Driving a car or other vehicle = strained eyes.
Using a computer for long periods = strained eyes.
Walking or looking into the sun, or not wearing sun glasses= strained eyes.
Not being aware that you are in need of spectacles and should go to an optician= strained eyes.
Using an old fashioned blackboard and chalk = strained eyes. (no longer used for that reason)
over using your eyes or not resting them = strained eyes.
Suffering from dry eyes = sore eyes.
Working in an hot or dry atmosphere environment = sore eyes.
Walking into a drying wind = sore eyes and also a mixture of eyes strain.
Trying to see in the dark = eyes strain leading to sore eyes.
That is the British perspective.
Where the cause determines which word to use and eye strain will of course lead to sore eyes.
So you can get sentences where both words are used DOCTOR -> TO Patient. "you need to rest your eyes they are very sore and you have been straining them at work on your computer".
"A sight for sore eyes" is still pretty commonly used in my opinion. I'm American and not an old person.
Sore eyes generally are result of an allergic reaction or infection. They are red, itchy, and/or irritated.
Strained eyes (I think you mean this not "eye strain" which refers to a condition) are when the eyes are acting like strained muscles. They could be tired, blurry, or even give you a headache.
I couldn’t agree more, Ryan; however, I’ve seen “I’ve got sore eyes“ in Miles Craven’s book. I mean Real listening and speaking 2. I’ve got baffled.
As you pointed out that the following sentence is wierd in American English
( I’ve got sore eyes)
but the following one is more prevalent
( I strain my eyes) or ( I suffer from eye strain)
so can I mention “I’ve got sore eyes” is common in BRE ?