I have a battery jump-starter for many years (more than 5). I wonder if I should replace it now. How can I tell if it needs to be replaced?
I would replace the whole thing. Not only because lead-acid gel cell batteries typically won't have a shelf life of more than 5 years, but also because today, a one-pound lithium ion jump starter can provide the required current and the 17 lb beast you have is needlessly heavy.
Your jump starter due to its weight almost certainly has gel cell lead-acid batteries (or probably just one). I had after 5 years of use a pair of gel-cells worn out in my UPS, which was nearly always connected to the mains, i.e. the batteries were continuously float-charged which supposedly should be more forgiving to the batteries than leaving them unused for many years (even though I did check the float charge voltage and adjusted it slightly down from 13.8 V to 13.65 V in the middle of the recommended 13.5V - 13.8V range for room temperature).
Fortunately, the newer lithium ion cells should have a better calendar life and less self-discharge in addition to the major advantage, less weight.