What then is the purpose of Bitcoin whenever they say "Bitcoin was born in the wake of the 2008 recession" or is it just a mere saying?
Bitcoin
literally was "born in the aftermath of the 2009 recession". The signs of a recession were present in 2008, but it wasn't until 2009 that it really hit, and this was the year when the genesis Bitcoin block was mined.
Some people believe this was a statement, that Bitcoin was meant to address the failings of the banking sector. Some others took this to mean that Bitcoin could somehow prevent a recession.
The true intentions of Satoshi Nakamoto are not public, but the white paper suggests that; it is to be a
trustless payment system which connects transactors directly, without the need for a third party. With no mention of it replacing the current financial system.
without saying, it is clear that the Bitcoin price movement largely mirrors the United States stock price movement.
There are also varied opinions on this, as some people believe that they are uncorrelated. I personally don't think there is enough data to show that Bitcoin mirrors the price of U.S. stocks.
Usually, the current global economic situation would influence people's decisions to invest in assets like Bitcoin. So a recession would likely reduce the number of investments, while a boom will leave people with more spare cash to gamble with. This is not peculiar to bitcoin.