DOGE is not a temporary agency. It wasn't set up as one. The type of employee category that the notorious outsiders occupy makes them temporary employees with a maximum stay in their position--dunno if there's a requirement that they not be rehired for some amount of time. But I've read nothing decreeing DOGE temporary.
DOGE is not unaccountable; it's accountable to an elected official, Trump, all more so because it's administratively/hierarchically housed in the Office of the President--not some other cabinet-level-headed agency, not some "independent agency" which sets policy but apparently is to some extent unaccountable without a long-ish process for removing the level of authority/accountability pass-through called the agency "head" or "chair" or whatever the office's title is. (Whether the chief executive exercises that role is a different matter. But this is true regardless of the administration at issue--Obama and the US Dig. Services, Biden and US DS, or Trump and US DS/DOGE.)
Musk is not DOGE; DOGE will continue beyond the legislated (or regulatory-agency imposed) limit to Musk's sojourn.
Then there's the question I haven't seen posed: If Trump thinks that having the executive power invested in him, will he accept limits on how he runs his own office. Note that I'm pleading ignorance here, I don't know if that "special temporary employee" limit was imposed by some regulatory agency under the authority of the President, whether Congress with Presidential agreement placed this limit on the Office of the President--and, if so, whether it was one of these "you're going to shut down the entire government" or "you're going to do this horrible, horrible thing" over something so piddling as the # of days a seldom-used "special temporary employee" can work in the OP?