It is possibly a legitimate email. As the other answers state, it's best to not click on links in emails, as even when you do have legitimate business with a bank you may get phishing emails that appear to come from them.
However, this at least passes the smell test as an email you expect to receive from a source you expect to receive (e)mail from. Synchrony Bank is the bank that handles Amazon store cards. They are required to send you a letter informing you of the reason of the denial, and while nearly all of these letters come in physical mail to ensure delivery (as they are required by law), it is permissible to send the notice via email, and if they did likely they would put the information behind a basic identity check such as DOB and SSN (the key identifiers in the banking world).
It is also possible that the card was denied due to a failure in the credit check; that's fairly common, and can be for a lot of reasons (you have no credit history, you have a freeze on your credit report, etc.). Banks want to offer you credit, so if the reason was something that you could fix, they might send you an email that offers the possibility to rectify the issues, beyond the full refusal letter. I've received this kind of email before, not from Synchrony but from similar organizations, and again they would gate the application site behind at least basic identity verification like DOB and SSN.
As such, it seems like you should not dismiss this email as clearly phishing, but instead take basic measures to ensure you go to a legitimate site. Call their support line, at 877-295-2080, or, perhaps quicker, visit the Amazon card site and click to Chat with an agent on that page.
And just to double-underline this: you shouldn't trust the number in this post either. Visit the sites directly to verify their authenticity, and even better, search them out yourself. For reference, to find these two pages, I searched "Amazon Synchrony" and "Synchrony Contact Us".
