What does it mean to allow an app to open supported Links?

It's really hard to answer that if we know exactly what the app is going to do.

It may open links, but that's about it. There are many, many apps that simply open web pages. It will be a bad decision if the link is not the "default" or a recommended browser. If it's the former, and the user uses another app to open it, Google will likely penalize it - they don't care about opening that other app. If the other app opens it automatically though, then you can allow the link to be opened there too (but no more than once every 30 days).

So this is a broad policy, but it's important for us to remember that it applies equally to all apps. You can't say "Google doesn't care about this one app, so it won't care about my app either."

You're essentially asking the same question again, but as phrased it shows you understand the issue far better. We actually gave an excellent answer to that before here In your case I think you're correct that the answer is no. At a minimum that would mean being a web browser and handling certain URLs the same way it handles them across all applications would make sense. We don't have a firm answer on how that would or wouldn't affect the policy in terms of what's considered to be opening a link, or the number of times you could use that kind of functionality.

Why can't I open Links in my email on my Samsung phone?

I am using the default email app on my Samsung Galaxy Note 5 and it worked perfectly, I was able to access links from email apps as usual.

However, all of the sudden now, after upgrading to Android 6.0 Marshmallow, the only way to open a link in an email is to open the mail client, tap on the link, wait for the link to open, then copy the URL. Is this a bug? Shouldn't the stock app be able to do what it was used to do? It's possible to change the settings so the phone can open emails directly in the email app, but if it's not broken, why is it broken?

Incase anybody else comes along with this problem. In my case, the "Open links in" setting was greyed out, indicating that the setting was not available. But in other words, it was turned off and the app was unable to open a link from an email.

The reason was that I was testing the app on a Google Nexus 6 (Android 6.0), whereas it was able to correctly open a link in an email on my Samsung Galaxy Note 5 (Android 6.0),
The conclusion - when you are testing your application on other devices than those used to test the app, try doing what the app developers say, which is to reboot the device in question.