It is true that as a term it has accrued some negative connotations due to the frequent use of the all-encompassing "regime" to speak of governments where their exact denomination tends to fall on the autocratic side of things. From a journalistic point of view, it is better to use a neutral term than a charged one; which unfortunately as you've noticed yourself it can taint the term to readers who are not familiar with its exact scope.
But it is correct to call it Biden's regime, just like the current administration (perhaps a better term given its popularity in the US) is part of Trump's regime.
https://en.wikipedia.org/wiki/Regime
Relevant quote: "In politics, a regime (also spelled régime) is a system of government that determines access to public office, and the extent of power held by officials. The two broad categories of regimes are democratic and autocratic."
It is true that as a term it has accrued some negative connotations due to the frequent use of the all-encompassing "regime" to speak of governments where their exact denomination tends to fall on the autocratic side of things. From a journalistic point of view, it is better to use a neutral term than a charged one; which unfortunately as you've noticed yourself it can taint the term to readers who are not familiar with its exact scope.
But it is correct to call it Biden's regime, just like the current administration (perhaps a better term given its popularity in the US) is part of Trump's regime.