is it totally meaningless or impossible to do factor analysis between two measures with different age ranges? For example, I want to test if two personality measures are testing the same underlying broader construct (like latent factors...), but the two measures are tested in different ages of the same group of people (age difference like 5-7 years), is it statistically meaningless to do so because I can not ignore the age difference between the two measures?
It's not statistically meaningless. It will tell you about the factor structure.
There may be interpretation difficulties - what if a factor emerges that is really just an artefact of age? For example, if you measured height and math ability in children, these would be likely to load on the same factor. But that's just because older kids are taller and better at math. If you did that factor analysis in adults, or in kids of one age group only, you'll get a different factor structure.
Similarly, the same question might mean something very different to people of different ages. "I like playing games" or "I like going to parties" mean very different things to a 5 year old, a 13 year old, a 20 year old and a 50 year old.
There are methods for assessing whether factor structures are consistent across groups - these are known as factorial invariance analysis. You can compare item intercepts, error variances, loadings and factor means and variances across groups. These are typically done using confirmatory factor analysis, but there are also approaches using exploratory factor analysis.