It can, actually. JPEG is just a codec. JPEG/EXIF and JPEG/JFIF are the file formats commonly used. There’s nothing stopping you from encoding an alpha channel and making a new JPEG/ file format. There isn’t a need to do so, so there isn’t a standard file format for it. Most use cases for transparency won’t work well with lossy images (blurred edges will be a big problem), so PNG or GIF is used instead.
And if you’re wondering, there is a new file format that does support alpha and is considerably better than JPEG: HEIF/HEIC
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