Michael Bloomberg was always a Democrat, a moderate authoritarian centrist Democrat like Lieberman, the Northeast type that has been dominant since 1992 when the country was realigned.
Bloomberg just ran as a Republican in 2001, 2005 and 2009 to get their votes because he was law and order and he doesn't like identity politics, which Democrats have been doing in NYC for years, especially with the rise of Al Sharpton.
Bloomberg is definitely more liberal than Lieberman, although that is not saying much. And Bloomberg ran as a Republican because it would've been impossible for him to have beaten Mark Green in the Democratic Primary (who was extremely popular with the Dem. base at the time and was widely believed to be the inevitable next mayor of NYC), not because he "didn't like identity politics." Bloomberg has been a Democrat for most of his life.
