Asking for Ginjo's body is an astonishingly bold move from Ichigo; a direct test of Soul Society's loyalty to him, in the same way that the fight with Ginjo was a test of his loyalty.
Ginjo was an enemy, it's true. But at one point he was also a deputy Soul Reaper...you know, the same thing Ichigo is now. And so by requesting Ginjo's body for burial, Ichigo's not just asking for a favor, he's asking them to show them that they do view him as an ally, by giving Ginjo the respect that Ichigo feels he himself is owed.
He didn't need to do this! I think practically speaking, Ichigo would have been fine under the status quo. But he wanted reassurance, a sign of commitment that the people he fought for would respect him as well. And I think he gains some of that respect here, just for asking.