Learn English – I am so confused by, for example, “People love their life or lives.”

I take the real question here to be: "Why do some people pedantically cling to dying forms?" That's a good question. I think the answer is relatively straightforward.

People who want to present an air of education and in general lay claim to upper class privilege are the ones who tend to do this. It's largely because it is an index of education and a high degree of literacy (either that or role playing games, which is somewhat different, but still primarily an upper middle class past time...).

There is a lot more to it than that, of course. There is an intricate set of language ideologies which give rise to this kind of behavior. But the short answer is that in using such forms people attempt to identify with culturally and economically powerful (hence linguistically conservative) groups .

Everyone does this to some degree, of course.

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.