Many individuals lost their individual life. or Many individuals list their individual lives.
Each person has one life right?
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In the simplest sentences, the object agrees in number with the subject.
However, the object does not need to agree with the number of the subject and the verb. None of these is incorrect:
In your sample sentences, the object life takes the plural pronoun their, and each of the sentences carries a different meaning.
You could also take a look at page 54 of Rodney Huddleston's English Grammar for some more details and examples.
In cases like this, and unlike some other languages, English tends to use what is sometimes called a "distributed plural", so even though each person only has one life, the plural is used as though representing "all the lives together". Similarly: "The teacher asked the pupils to get their notebooks out".
"Many lost their lives" is correct. "Many" is plural, "their" is plural, "lives" must be plural too.