Birth registration basically provides documentation of you as a legal person, so it has implications not only in the early stages but also for the rest of your life. For example, it prevents people from registering for school, getting access to health care and registering to vote, so civil and political rights as well as economic and social rights are all implicated in this.
I would also say that in the resettlement context, if a child does not have a birth certificate, then UNHCR basically has to try to approximate a birth date. While this is helpful because it gives people the ability to be resettled, the problem is that often these birth dates can be off, because you have to rely on people remembering some kind of significant political or natural event, for example. This can have consequences for children being placed in the wrong grade.
At the other end of the spectrum, it has consequences for when people are eligible to retire because, in our society, everything is based on chronological age. If that chronological age is off by a few years, it can have repercussions across the life-course.