The second is that the claim that "actually nothing was done for a long period of time" is wrong, and the claim that Obama didn't declare a national health emergency is very misleading. The Centers for Disease Control on April 24, 2009, declared that the containment of swine flu in the United States was "not very likely." Two days later, on April 26, the Obama administration declared a public health emergency, before any Americans had been killed by the swine flu - the same step Trump has now taken. A couple days later, Obama asked for and received billions of dollars from Congress to deal with the swine flu.
Obama didn't declare a more drastic national health emergency until six months later, but Trump hasn't done that either. And that brings up the other key point here: The reason for the less drastic initial response was because the swine flu was significantly less dangerous than today's coronavirus.
While it's true that more than 1,000 people had died and 20,000 had been hospitalized by the time Obama declared a national health emergency, the fact that those number occurred over the course of six months is telling. The spread of the disease was much slower and the mortality rate was much lower than it is today. At the current rate, the United States would hit 20,000 cases in a matter of weeks, not six months. What's more, the swine flu had a mortality rate of 0.02% - about one-50th of the lowest rate health officials are citing for coronavirus today (1%).