It seems I want to convert audios, which I want to stream on my website, to audio/mp4; codecs="mp4a.40.2".
Using ffmpeg-cli-wrapper, I am conve
In your ffmpeg command, use the extension .m4a (or .mp4), not .aac:
ffmpeg -i /tmp/input.any -acodec aac -b:a 256000 /tmp/output.m4a
The .aac extension is used for AAC in the ADTS container, which is not a format that is widely used or supported on the web. The .m4a or .mp4 extension is used for AAC in the MP4 container, which is far more common.
Or, if for some reason you really want the file to be named output.aac but yet want it to contain AAC in MP4, then you can specify the output format instead of having ffmpeg guess the format that you want based on the output file extension:
ffmpeg -i /tmp/input.any -acodec aac -b:a 256000 -f mp4 /tmp/output.aac
But if you use a non-standard extension like that then you may also need to update your web server configuration so that it will report the correct media type.