But not really right? It could happen in the market, company A chooses to bailout company B by buying it and investing money to keep it afloat.

Except company A in this case is the government. No? Why is it that when it is the government doing this action, it has to gift the money instead of potentially profiting from it?

Edit : just saw the edit. I see! But still, I don't get in what sense it is more socialist, instead of just saving the companing 'for free', people actually buy (forcefully invest?) in the company to save it. If anything it makes it more capitalist if the taxpayers invest in the bailout, instead of just giving it away!

Its pure propaganda playing on American fear of socialism.

In other very capitalist economies governments did take stakes in banks in return for bailouts. The first British bank that needed one (Northern Rock) was entirely taken over by the government and shareholders just lost their money. The government bought stakes in others. It was still criticised as being too generous to shareholders and management.