There could probably have be thesis written on this question. But let me assemble a few words I know and make an attempt.
The Buddha, was also known as Shakyamuni, or Gautama. Both names signified where he came from. Although none of these terms state any royal connections, though do signify aristocracy. Shakya was the name of his clan, and Gautama the name of his family lineage. Thus, some texts even depict his mother’s name to be Gautami.
Nevertheless, he did belong to an family of the ruling class. He had access to luxuries, equivalent to that of royalty. Since he fell in the category of the ruling class, where power lies, he was at the centre of attachments and worldly pleasures that his father (Shuddhodhana) could offer him. Siddhartha was kept engaged in opulence of culinary creations, never ending wine, and dancers. It stemmed from the prediction, that he would either be a great sage, or a Chakravartin Samrat aka wheel turning ruler. Because his father did not want Siddhartha’s heart to melt at any tragedy, they were kept out of this site.
Even if he was/was not a Prince, either ways nothing much changes. His aristocratic family would place him a position of responsibility, such that he would be expected to carry on family name and leadership. They would in both scenarios, look at him as someone who would take the seat of power even further. Thus, Siddhartha was protected from tragedies of life around him.
But the Buddha’s life story further asks the question (in title), when he comes to mainland India from Shakya (Kapilavastu). In India then, was when he is called a Kshatriya. This is a reason for us to believe he may have been a Prince. Here, the people and the texts written in reference to mainland India (today as well) refer to him as a Kshatriya. Kshatriya is person of the fighter clan; this community of people could mostly be royalty in those days. However, it is possible that people interpreted his (renounced) position of power, as him belonging to the Kshatriya cast. So, he may or may not have been be royalty, but for the Vedic preachers in India, would be perceived as a Kshatriya.
Another point of confusion to note here is, that it may not be wrong to merely perceive him as a Kshatriya during that time. Since his family belonged to a seat of ruling power, and included fighters. However, in order to recognise yourself as a Kshatriya, one would have to recognise the Varna system, which The Buddha sincerely denounced.
So it may not be have been wrong to call him a Prince, or not one. But it would be wrong to call him a Kshatriya, since he did not follow the Vedas, and the Buddhist texts stand opposite to many core ideas of the Vedas.