Browser update

Excerpted from Update to the latest version of Safari - Apple Support:

If a website says that your browser is out of date

If a website says that Safari is out of date even though you're already using the latest version of macOS or the latest version of iOS or iPadOS, there could be an issue with the website. If you’re sure that you want to use the website, contact the website owner or developer.

Websites that implement a "browser check" incorrectly need to be fixed. Only the website owner or developer can do that.

Safari is updated through Software Update (apple menu/system settings/general/software update), not through the App Store. If there is an update available, you can find it there. If there is an OS update available for your Mac model, Safari may be updated within that.

Some websites have features that require, or work better with, newer versions of your browser. I had this issue when I was using an older MacBook and couldn't update Safari anymore. I was able to use up-to-date versions of FireFox for quite awhile, and that may solve the issue for you.