Yes, you need to update TurboTax in order to use it.
I'm not sure why your computer doesn't recognize Intuit as a software publisher, but you need to give permission to update, or you won't be able to use the software.
How do I know it is not a scam or data breach to get my information?
This is a very important concern that Intuit isn't addressing. They need to identify their update as coming from Intuit and register it properly. With something as sensitive as taxes and bank accounts for direct deposit, the answer to just update or it won't work is totally unacceptable.
This isn't an isolated event. Many people are having this problem and many more just quickly click on the install even though there is an unknown publisher warning. You need to get your software developers to properly identify the update package, otherwise you are encouraging people to follow very unsafe personal security, something your company should be very concerned about, by installing software of unknown origin.
Please call TurboTax Customer service. Here is a link: Turbo Tax Customer Service