The poem explores a prince who is ostensibly powerful and important, but completely constrained (both mentally and practically) by his surroundings to fulfill his specific role; even if he grows up to be aware of this he will be powerless to change anything.
This is shown in several ways. In the first line he's said to get anything he needs, not wants; implying somebody is making the decisions for him. Then a list of fairly fancy sweets to establish his wealth, plus "bubble gum" - this is to imply that the reason he is given this is mainly to placate him and not to signal wealth or teach good taste, and if something is effective in doing that it doesn't matter too much if it's low-status.
The second verse starts with him being served ice cream while studying, connecting the verses and reinforcing the previous point. It also hammers in that his education is limiting what he knows about, and that he is not free to explore topics that interests him.
The third verse escalates the heavy-handed control; that his "friends" are "lined up" and he's given an hour to play with them strongly implies that they aren't his friends at all, but employed by the court as a sort of simulacrum of a real childhood. That his clothing is described as a "uniform" and not e.g. "attire" is to hint at the royalty is not absolutist, but perhaps a figurehead monarchy beneath a military junta or the like. It also serves the purpose of keeping the time period fairly modern, so the reader won't imagine a medieval setting.
As his clothing won't get stained, it means he's not really participating in the games he runs, but mostly ordering the other kids around. There is also no hint that he's rebelling against this, but that it's seen as a natural part of his life.
The fourth verse is yet another escalation; this time describing how his subordinates are controlled by basically keeping their families hostage, implying a political situation similar to North Korea, and that they would have an incentive to harm him otherwise. That he "wakes assured" from this shows that he's either unaware or accepting of this.
All the verses up until now has followed a similar pattern; an escalation of some mood in each line, followed by a drop or reversal of it in the last one. The verses themselves follow the same pattern, and now in the fifth one we get a more somber bird's-eye view of his future. That his garden is purged of birds, pests, and weeds (perhaps somewhat obviously) symbolizes his life, and that it states that he will die here implies it's inevitable and nothing will change. Moreover, the rhymes repeat here, with "weeds/leads" connecting back to "needs/seeds" in the first verse. This implies an unbroken cycle, that there will be another prince after this going through the same motions again.
Finally you might analyze the prince's situation as symbolizing current society in some way. This would be unintended from my part; the poem was intended to impart a bleak particular image of an imagined individual and society, and not be intentional social commentary.