The message has nothing to do with TurboTax, it is how Apple choose to treat ALL  applications not purchased from the Apple Store - all non Apple software will get that message with OSX 10.15 (Catalina).

See this Apple article: https://support.apple.com/en-us/HT202491

And:  https://ttlc.intuit.com/community/troubleshooting/help/turbotax-for-mac-won-t-open-when-installed/00...

https://ttlc.intuit.com/questions/2899293-turbotax-for-mac-won-t-open-when-installed

https://ttlc.intuit.com/questions/1899267-administrator-rights-mac

If I can't use this product will I receive a refund? 

Yes. Please click the link below for our TurboTax Satisfaction Guaranteed refund policy.

Our TurboTax Satisfaction Guaranteed refund policy - Community