"Many lost their life" or "Many lost their lives"

Many individuals lost their individual life. or Many individuals list their individual lives.

Each person has one life right?

In the simplest sentences, the object agrees in number with the subject.

• He is a student.
• They are students.

However, the object does not need to agree with the number of the subject and the verb. None of these is incorrect:

• Most families today own a car.
• Both of them sprained an ankle during the trek.
• They all thought they had an answer to the problem.
• Teenage vandals are a problem in this neighbourhood.

In your sample sentences, the object life takes the plural pronoun their, and each of the sentences carries a different meaning.

• Many lost their life. (All of them together had one life to lose: their life.)
• Many lost their lives. (Each of them lost one or more lives, practically understood to mean that each of them lost their own life as people usually have only the one life to lose.)
• All my opponents lost a life trying to collect that torque bow in Level 7 of the game.
• All my opponents lost lives trying to collect that torque bow in Level 7 of the game.

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".