You can only download the tax program to 1 computer, but you can prepare up to 5 different tax returns using the program on that computer. Hope this clarifies your misunderstanding.
Yes you can install it on 5 computers you own. You can do unlimited returns and efile 5 for free. The rest you can print and mail. To install it on another computer you need to log into your account, all your tax programs will be listed so you can download it to another computer, even get the other Mac or Windows.
Since you now have to log into an Intuit Account when you first install it, all your downloads and licenses are listed there (even if you bought it from somewhere else) so if you need to install it again or install on another computer, even on the other operating system (Windows, Mac) you can download it again.
How to get started with the Desktop program, install and activate it
https://ttlc.intuit.com/turbotax-support/en-us/help-article/product-delivery/download-software-turbotax-com/L7u9oLEkq_US_en_US?uid=luqgvzrd
Access your downloads here
https://shop.turbotax.intuit.com/commerce/account/secure/login.jsp?pagecontext=download
Here is mine for 2023