EDITED: What JimmyM said
Not strings but the recording or your tuner. The fact that you can tune to the recording and everything is fine means your strings are good enough to tune. What you are experiencing is the same thing as going in to play with someone on an out of tune piano since they cant tune to the bass the bass must tune to them. It is nicer to play with something that is in tune because it helps you to train your ear but you don't always have a choice. Check your tuner against another tuner or a tuning fork just in case it is somehow defective.
I have seen old strings become hard to intonate at the upper frets (requiring a bridge adjustment) and loose some harmonics but you are OK for now. I have used a single set of strings for about ten years. Don't know if I would do that again but it is possible as long as they are not corroded and kept reasonably clean.
Peace,
S