Yes, these drugs can damage your liver but they don’t do that to everybody. I don’t know about the kidneys (so far as I know, mine are healthy) but it sounds like your kidneys are being properly monitored anyway.
See here for general information about liver tests: https://en.wikipedia.org/wiki/Liver_function_tests Maybe you are already being tested regularly but you didn’t realize it. The most basic tests are GGT, AST and ALT. Febuxostat is known to increase ALT in particular.
There are many causes for high uric acid.
There are tests you can do such as measuring the total amount of uric acid in your urine which could help with determining why your uric acid is high. It could indeed be a problem with your kidneys which have trouble getting rid of the stuff but your body could also simply produce way too much uric acid when you’re not taking medication.
It sounds like 300mg allopurinol would be sufficient for you in the long run but one problem is that you may still have uric acid crystals in some parts of your body. Blood tests do not detect them and they can cause your uric acid to spike back up in between tests and possibly lead to kidney stones in spite of your low uric acid tests. These crystals should go away slowly if you keep taking medication and might be a reason to keep your uric acid unusually low for a while.
So considering you had a stone in spite of 300mg allopurinol (have I got that right?), maybe 40mg febuxostat would be a bit too low. If you can get the 120mg pill, perhaps you could split it in half to get 60mg for a while and then try switching to 40mg later. You could also try to get around 60mg (precision is not important) from 80mg pills with a pill cutter or other tool. Allopurinol is easier to fine tune because there are 100mg pills everywhere while many countries do not have low-dose febuxostat pills.
Exercise, doing drugs such as alcohol during the weekend and many other random variations in your diet and lifestyle could also raise your uric acid in between tests and possibly contribute to the formation of stones. Having very low uric acid test results would provide a buffer against all sorts of changes.
To be clear: I’m no doctor and have had no kidney stones. I’m only making guesses here based on my experience with these drugs, what I’ve been told by my doctors and so forth. I can’t provide you a medical opinion on what your uric acid target should be.