Chlorine is added on purpose, and it is supposed to still be there.
What it is
A utility disinfects at the plant and then has to hold a measurable residual all the way to the last tap on the street, so chlorine arriving at a house is the system working as designed. Many utilities now use chloramine, which is chlorine combined with ammonia, because it holds longer in the mains. The two behave differently in treatment, so it is worth knowing which one the local utility uses.
What it usually does
At household levels this is usually a taste and smell complaint, strongest in homes near the plant and after hydrant flushing. Chlorine also fades if a glass stands for an hour, while chloramine mostly does not. Both are hard on rubber seals and on aquarium fish, and both will shorten the life of softener resin over years.
What it does not prove
A chlorine taste does not indicate a problem with the water, and removing the taste does not make the water better in any other way. Chlorine residual also says nothing about lead, which in city water usually comes from the service line and the household plumbing rather than from the plant.
When to test
Usually not worth a lab test, since the utility publishes an annual report with the numbers in it and will say whether it uses chloramine. Worth a reading only when sizing carbon, or when a specific use such as an aquarium or a dialysis machine needs the residual at zero.
See what this usually means for your water.
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