"Designation" is a series of questions about what a name really means. The speaker starts with the practical view: a name is something you own, and it lets a neighbor call for you. Then they push further. If a name is only a label stuck on by other labels, why should anyone care about it?
The questions get more personal as the poem goes on. By the fourth stanza the speaker is asking why they should keep their own name, which they say is tied to nothing but themselves. That "self" turns out to be a tired, "haggard child" in a dim room the speaker often forgets. The name seems to carry a memory they would rather leave behind, or at least feel distant from.
The mood is weary and quietly bitter. Nothing is shouted. The closing "Why, indeed" works like a shrug, leaving the question open and the speaker unsure whether an answer exists.