How to output from TileMill in a format like openstreetmap.org standard

TileMill uses CartoCSS as a styling language, but the actual OSM.org Standard style is written in Mapnik XML and thus can't be used in TileMill directly. It is technically be possible to create a TileMill project that closely approximates the standard style, but as far as I know no one has created such a thing.

If you want tiles that look exactly like or very similar to those on OSM.org, it may be easiest to use something like generate_tiles.py with the standard Mapnik style. Otherwise you should see if playing around with the colours and style of the OSM Bright template (which it sounds like you already have set up?) to get close to what you want.

OpenStreetMap uses a CartoCSS-based stylesheet since August 2013: openstreetmap-carto. So you can load that directly into TileMill directly. You can then edit it as you wish, export a .xml stylesheet from TileMill, and use that with your main renderer.

If the machine that you use for rendering isn't the same as the one that you have TileMill installed you may need to check for renderd errors in its logfile (missing fonts, file paths not qualified, etc.) after transferring the stylesheet - files exported by TileMill may refer to components of the original TileMill project, and in turn (at least with OSMBright, but not with openstreetmap-carto) there may be references to TileMill's download cache.