This issue report complains that Edge doesn’t stream AAC files and instead tries to download them. It notes that, in contrast, URLs that point to MP3s result in a simple audio player loading inside the browser.
Edge has always supported AAC so what’s going on?
The issue here isn’t about AAC, per-se; it’s instead about whether or not the browser, upon direct navigation to an audio stream, will accommodate that by generating a wrapper HTML page with an <audio> element pointed at that audio stream URL.
A site that wants to play streaming AAC in Edge (or, frankly, any media type, for any browser) should consider creating a HTML page with an appropriate Audio or Video element pointed at the stream.
The list of audio types for which Edge will automatically generate a wrapper page does not include AAC:
audio/mp4, audio/x-m4a, audio/mp3, audio/x-mp3, audio/mpeg, audio/mpeg3, audio/x-mpeg, audio/wav, audio/wave, audio/x-wav, audio/vnd.wave, audio/3gpp, audio/3gpp2
In contrast, Chrome creates the MediaDocument page for a broader set of known audio types:
static const char* const kStandardAudioTypes[] = { "audio/aac", "audio/aiff", "audio/amr", "audio/basic", "audio/flac", "audio/midi", "audio/mp3", "audio/mp4", "audio/mpeg", "audio/mpeg3", "audio/ogg", "audio/vorbis", "audio/wav", "audio/webm", "audio/x-m4a", "audio/x-ms-wma", "audio/vnd.rn-realaudio", "audio/vnd.wave"};
If the the response sends Content-Type: application/octet-stream, includes a Content-Dispostion: attachment, or puts a download attribute on the anchor <a> element that leads to the media, Edge will download the media file instead of playing it in the browser.
Note: In Windows 10 RS5, the extension model is capable enough that it’s possible to write a browser extension that intercepts navigation directly to audio/video Media types and renavigates to a wrapper page. [Sample code]
-Eric
PS: Edge has similar special handling for video types:
"application/mp4","video/mp4","video/x-m4v","video/3gpp", "video/3gpp2","video/quicktime"
text/plain 
A great test page for audio codec support in a browser is this page https://hpr.dogphilosophy.net/test/
I’m very happy to see that Edge 17 handles opus just fine (or is it Edge 44, the numbering is confusing, CanIUse.com uses 17, Wikipedia says 44), by that I mean inside a ogg container, can’t recall which version it was added, Opus was supported before with webm though.
BTW! Edge 17 (44!) passes all tests on that test page besides CAF, but that is a is not much of a surprise I guess. In my future projects I’ll now use Opus audio files (possibly with lower quality MP3 as a fallback) now that all major browsers support it (Safari is lagging behind though).