Not only was Obama not inaugurated until 2009, but the Great Recession began in 2007, so tying it to Obama because it is popularly associated with 2008 and Obama was elected towards the end of that year is...doubly wrong.

I'm also certain some folks believe Biden was President during the start of COVID-19 in the US too. (Jan 2020.)

It seems to be a pattern, where a disturbingly-large minority of people remember nothing but the year-number of an election (e.g. "2008") and never account for how voting was at the end of that year and the voting-result only takes effect the next (odd-numbered) year.

Oh, there's also a general "I want it to be true so I remember it that way" aspect, but I think that's contributory rather than the main cause.