The question gives the example sentences:
(1) People often make a lot of mistakes in their live.
(2) People often make a lot of mistakes in their lives.
Sentence (1) is clearly incorrect in that the singular of "lives" is "life", not "live". The word "live" is the base form of the verb "to live". The word can also be an adjective or adverb, but not a noun. See Merriam-Webste. So let us correct (1) to:
(1A) People often make a lot of mistakes in their life.
In my view (1A) is also incorrect, or at least unnatural. As this Google Ngram shows, the plural form of this construction is far more common, at least in the Google corpus.
Beyond Google, I feel that (1A) would imply a single shared life for all the people under discussion. That is, of course not the case. And while a listener or reader would know that is not the case, I think the form would strike most fluent speakrs as wrong, or at least odd and unusual. It would break the flow, and distract from the author's point. Form (2), on the other hand, is natural, and would not draw special attention.
This is part of a more general situation in which ther is a group noun followed by a related countable noun. In most cases the countable noun is typically used in the plural form, but in some cases the singular has become customary in English. Some examples:
- Most customers will draw on their accounts.
- Most machines have off switches,
- Most machines have an off switch.
- Most employees spend several hours a day at their jobs.
- Deciduous trees lose their leaves in the fall.
- Balkan countries have been noted for their wars.
- Programmers stereotypically drink cola and other caffeinated beverages.
- A troop is only as good as its riders.
- An organization depends on its members.
- Many police wear uniforms.
- Many police wear a uniform.
There is no clear rule that applies to each such case. Usage varies by the noun involved