Changing cape names is primarily a PR tactic, not a legal tactic.
When a parahuman changes their cape name, the goal is to separate the two identities in the public mind, rather than accomplish any sort of legal shenanigans.
Case 1: If the public is not aware of the change, then the cape can present their new name as a completely new parahuman, essentially jettisoning any bad reputation associated with their old name. From your list:
Madcap => Assault: a villain disappears and the PRT gets a shiny new hero with no inconvenient questions asked
Hijack => Regent: a dangerous child of Heartbreaker disappears, and a new (much less dangerous-seeming) minor villain appears elsewhere
Mockshow => Romp: pretty much the same as Madcap/Assault
Harbinger => Number Man: nobody would knowingly use a former member of the Slaughterhouse Nine as a banker
Case 2: If the public is aware of the change, then it typically represents a way for the parahuman in question to renounce their previous (mis)deeds and attempt a reboot of their public perception. Essentially, the name change is intended to signify to the public that the cape has changed as a person; in villain => hero cases, this can represent a way to say, "I don't want to be that person any more." From your list:
Sphere => Mannequin: It's not clear to what extent Mannequin chose his new name. It's possible that this one was not a voluntary name change but rather the public simply referring to him differently to try to avoid acknowledging the loss of so much potential.
Skitter => Weaver: Taylor is effectively abandoning her previous "scary warlord" persona and attempting to redefine herself as a dedicated opponent of S-class threats.
Glaistig Uaine => Valkyrie: Much like Taylor, she's trying to convey that her way of viewing and interacting with the world has changed.
A few name changes don't fit quite so neatly into those categories.
Armsmaster => Defiant: It's important here to separate what the public knows versus what the PRT and Protectorate know. Elements of the PRT and Protectorate are aware that Armsmaster broke the Endbringer Truce and was essentially sentenced to house arrest for the rest of his life. To help preserve the Truce, however, the public was not informed that Armsmaster had violated the Truce (which is why he got house arrest instead of the Birdcage); the implication seems to be that he simply retired. As seen in Triumph's interlude, the name change to Defiant lets the PRT conveniently pretend that Armsmaster continues to receive his punishment while still accepting the aid of a shiny new Tinker in hunting the Slaughterhouse Nine. From the public's perspective, a number of people appear to have figured out that Armsmaster = Defiant, and presumably write off that change as reflecting a change in operating style (Brockton Bay Protectorate leader => Dragon's S-class hunting companion). Basically, the PRT pretends it's case 1 for their own benefit, while the public generally perceives it as a minor version of case 2.
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Flechette => Foil: This one appears to be a variation of case 1. The PRT would have a major PR problem if it was known that one of their Wards ran off to join a villain group, so the name change appears to be a sort of peace offering from the Undersiders to the PRT to help maintain their secret truce.