Are you required to claim ownership of the site by your company in the footer or the about page?
No, in most cases you are not required to do so. Some countries have laws that mandate this, fx. in Germany a "impressum" may be required. You should see a lawyer in your area for a more precise answer.
Common wisdom in the micro-ISV community is that you have to earn trust before a sale, and this involves being completely transparent about who you are. If you don't your sales will suffer dramatically.
And if not, how does anyone else can legally find out if the site is owned by you personally or your company.(taxes/lawsuits etc)
Very easy, look up the domain ownership in WHOIS. All DNS records are public and show ownership information. Additionally, if you're selling something, follow the money trail to the recipients bank account -- most credit card issuers and banks will happily tell who owns a merchant account.
You are not anonymous on the Internet, certainly not when people bother to actually investigate things. Domain ownership is registered, your web host will most likely have logs of administrative access to your website, these logs have IP addresses registered, IP addresses can generally be followed to a household ADSL / cable modem, and so forth.