The quick answer to your question is that the concept of the Inner Child does have a scientific backing, though not in a traditional sense when following the deliniation laid down by Karl Popper (1959).
When looking at the science behind the concept, you need to unpick it from the pseudoscience put across in many books such as the one mentioned in the Wikipedia article you linked — Lucia Capacchione's Recovery of Your Inner Child (Capacchione, 1991).
The fact pointed out in the book that its roots are within ancient mythology and fairy tales is a little weak. However, the links made later within the same part of the book are more accurate in that they come from Neo-Freudian psychologist Carl Jung, along with Eric Berne's Transactional Analysis ("nurturing parent", "protective parent", "critical parent" etc. in the Parent, Adult, and Child ego states) and Sigmund Freud himself (see: ego-defense mechanisms).
Therories from Eric Berne, Sigmund Freud and Carl Jung are all often dismissed as pseudoscientific because they do not meet the falisifiability criteria within Karl Popper's demarcation between science and pseudoscience (Popper, 1959; McLeod, 2020). However, Sven Ove Hansson (1996; 2008; 2013) points out that Popper's criteria is too tight and excludes legitimate sciences. — See also my answer to the question on pseudoscience within Freudian and Neo-Freudian theories for more information on this.
When reading the synopsis from the jacket of Whitney Hugh Missildine's book Your Inner Child of the Past (Missildine, 1963), notice the sections of text marked by me in bold below:
What was your childhood like? Were your parents always making you do things? Criticizing you? Or did you always get your own way? Were you often punished as a child? Did your parents continually fuss at you with anxious reminders and directions? Did you feel neglected, even unwanted? Or were you the boss of the family for whom all sacrifices were made?
This remarkable book helps you to free yourself from the tensions that can be set up in childhood (even by parents who try hard to do their best) and continue to exhaust you in adult life. By describing — and illustrating with case histories — various child-parent relationships and their results, Dr. Missildine shows you how to recall the forgotten child you once were. He demonstrates how that child — who might have been overdisciplined or overindulged or subjected to any of a dozen common parental attitudes — still persists in the adult You.
You see how your 'inner child of the past' continues to act out old habits, old angers, fears and confusions. That 'child' today makes you, perhaps, oversensitive, or unable to relate to people, or subject to rages or procrastination or compulsive spending or being overcritical of others (or of yourself), and in other ways causes you pain or disturbs your marriage or family or working life.
To help you make peace with this 'inner child' and thus free your energies for adult living is the author's purpose. His concept of retraining the 'inner child' has led to happier, more productive lives for many of his own patients. In this book he speaks so clearly and interestingly, and with such warmth and sympathy, that you will find yourself relaxed and eager to begin discovering and coming to terms with your inner child of the past.
Using case histories bolsters claims made and provides the evidential science behind the theory. Maybe not the hard evidence purists around may be looking for (scientific equipment measuring something as evidence of existence for example), but evidence nevertheless.
When the Wikipedia article mentions that IFS therapy (Internal Family Systems therapy)...
calls wounded inner child sub-personalities "exiles" because they tend to be excluded from waking thought in order to avoid/defend against the pain carried in those memories. IFS therapy has a method that aims to gain safe access to a person's exiles, witnessing the stories of their origins in childhood, and healing them.
This is coming from the Freudian concept of ego-defense mechanisms of repression, and possibly denial. Detachment occurs often in abuse, rape and assault, and if occurred in childhood, the sense of "loss of childhood" can come into play.
There are issues with inner child therapy in these cases which I discussed in my answer to Can people improve their memory by training themselves to recall previously forgotten memories?, as trying and recall the whole memory can but not always lead to a change in the memories of what actually happened (False Memory Syndrome). However, if fully trained in this delicate area of inner child therapy and it is carefuly managed, it can be beneficial.
Capacchione, L. (1991). Recovery of your inner child: The highly acclaimed method for liberating your inner self. New York: Simon & Schuster.
Hansson, S. O. (1996). Defining Pseudoscience. Philosophia Naturalis, 33(1): pp. 169—176
Still to find DOI/PMID reference
Hansson, S. O. (2008). Science and Pseudo-Science. The Stanford Encyclopedia of Philosophy (Summer 2017 Edition) [Online]
Available at: https://plato.stanford.edu/archives/sum2017/entries/pseudo-science
Hansson, S. O. (2013). Defining Pseudoscience and Science. In M. Pigliucci, M. Boudry (Eds.) Philosophy of Pseudoscience: Reconsidering the Demarcation Problem(pp. 61—77). Chicago: University of Chicago Press.
McLeod, S. A. (2020). Karl popper - theory of falsification. Retrieved from https://www.simplypsychology.org/behaviorism.html
Missildine, W. H. (1963). Your Inner Child of the Past. New York: Simon & Schuster.
Popper, K. R. (1959) The Logic of Scientific Discovery. New York: Routledge.
Other source links
Ego-defense mechanisms (Simply Psychology)
Transactional Analysis (Wikipedia)