I check our chains every time I lube them, and we typically get 3500 miles on out 9 speed touring bikes, slightly more on our road bikes.
I keep them clean and replace them when needed. My wife's new touring bike is approaching 8,000 miles and the cassette still is looking good. Usually there is no need to change the cassette with every new chain. One of my road bikes, my all weather commuter, has more than 15,000 miles on the cassette, and it is still OK.
IMO-Chain/cassette/chainring longevity depend a lot on keeping them clean; the type of riding conditions, e.g., rain, sanded roads; loaded touring vs bare bike; etc.
PowerPoint slides from my presentations on bike maintenance at a wellness conference:
A clean drive train is essential to a well performing bike. This is the cassette that I mentioned above which was used most working day for 7 years. 15,000 + miles
This is one of the reasons it looks so good.
Measuring chain stretch.
Distance between links should be 1/2 inch.
Use a good machinist rule ( I use Park tool, it is easier to use) to measure one foot of chain. 1/16 inch stretch in 1 foot is OK, but time to think about changing chain. One eighth of an inch is definitely maximum stretch. When the chain stretch is between 1/16 and 1/8 inch, it is time to change chains.