Is the following combined sentence grammatically correct?1. Such modern examples recall an earlier time. 2. A name was a real part of one's...

The second combination is preferable to the first
combination of the three simple sentences.  But, perhaps you may wish to revise this
second version with the addition of the word and after the comma. 
For example,

Such modern examples
recall an earlier time when a name was a real part of one's identity, and not just an
identity label.

If it is possible
for you to add words for clarity, you may want to rephrase  the last part of the
sentence as in this example:

Such
modern examples recall an earlier time when a name was a real part of one's identity,
and not just a label to identify
people.