What this quote means is that there is (or at least there can be) a difference between what we do to make the money we need to live and what we do to enjoy our lives. Some people are lucky enough that there is little or no difference between the two, but many are not.
“Making a living” refers to the things that we do in order to get the money we need. There are many people who have jobs that they do not love. For them, going to work is something that you do because you have to, not because you enjoy it.
“Having a life” refers to having things that you enjoy doing and that make life a pleasure. For some people, this can include work. These people will have high motivation at work because they like what they do. They might well identify themselves with their work. For other people, “having a life” might refer to spending time with their families. For them, the enjoyable part of life only begins when work is over. These people will identify themselves with their families and not with their work.
In short, many people experience a disconnect between what they have to do in order to survive and what they want to do in order to have a fulfilling and enjoyable life.