Short answer; We don't know.
This is a far more complex question than it looks, but let me at least break down what I see as the major points to how to answer it.
In behavioural science it is generally thought that there are two primary contributors to behaviour and personality. Genetic (nature) and Environment (nurture). By taking out one of those variables and saying that the only consideration is environment, one could argue that the answer is yes; if the genetics AND the environment is the same, then so will be the person.
The problem I have with that is that it assumes that free will is a lie.
This is where I see that we miss a third pillar in our understanding of behavioural science; choice. Do we have the capacity to extend beyond our genes AND our environment and make choices to go in a direction different from both of these? The answer to this, and to what degree if the answer is yes, is essentially the part of the equation we don't know. That said, I'll spell out the cases for and against below. Let's start with the Against argument - that free will is not a contributor to personality and behaviour.
Stanford Prison Experiment
There is a lot of material available on line about this behavioural sciences experiment from 1973 and what happened during its execution. On the surface, it seemed to imply that force of personality and personal choice, in people who should have had an ample reserve of both, instead acted according to their environment which fundamentally changed their personalities during the course of the experiment.
This is the primary argument for the YES case; if you maintain the same environment, you'll make the same choices and be the same person because fundamentally, you are driven to do so. Of course, the implications are that this is that what we call free will or choice is really just a conditioned response to the environment and therefore we are all victims of the environment, not our own choices.
On the For side of free will however, there are other studies to discuss.
A behavioural sciences researcher has conducted experiments on cloned mice, putting each in an isolated but identical environment, then watching how they explore their environment. It seems that they develop very different characteristics in their exploration 'personality' and that they end up being very different creatures more often than not. This goes in direct contravention of the experiment cited above, but more, it is counter-intuitive in that if anything, you'd expect the mice to be identical (starting from scratch, with no manifest personality to intrude on their choices or free will) and the prison experiment to fail due to force of personality thanks to the established minds of those in the experiment.
So, what is happening?
We honestly don't know, but the thought may well be that free will finds it harder to intrude on minds already established with many patterns to fall back on and which make every new stimulus a smaller percentage of the world view in which it is to be integrated. This could explain the concept of old people 'being set in their ways' and the innate curiosity we see in babies, where their sense of wonder is enhanced by every new stimulus being a much larger proportion of their existing world view.
In your example, this kind of thinking means that your experiment is likely to fail on grounds that you are starting from scratch. That means that the baby clone has the ability to explore the early stages of his or her environment in different ways, and the early but minor differences then intrude on your clone's personality in manifestly fundamental ways later down the track.
I think that your better option would be memory transplantation (if you can develop or handwave the tech for it) as that would establish a strong body of patterns in the mind that are already learned and manifest. Starting at that later point, identical environments are more likely to have the desired effect as each new discovery is such a smaller component of the overall base of knowledge and ideas stored in the mind, meaning that each variance from your own experience has less impact on the now well established personality matrix.