For example, is it possible for banks like Bank of America to become a shareholder of another commercial company, say Facebook.
Sure they can (most publicly traded banks at least) - and they do it a lot.
Many banks have a proprietary trading desk, or Prop desk, where traders are buying and selling shares of publicly traded companies on behalf of the bank, with the bank's own money.
This is as opposed to regular trading desks where the banks trade on behalf of their customers.
There is no country tag, so I will answer the question generally.
Is it possible...?
Yes, it's possible and common.
Is it wise?
Ask Barings Bank whether it's a good idea to allow speculative investing.