That is not supported as you can see by having a look at the "Synopsis" part of the documentation of each GDAL tool. For example https://gdal.org/programs/gdal_translate.html
gdal_translate [--help-general]
[-ot {Byte/Int16/UInt16/UInt32/Int32/UInt64/Int64/Float32/Float64/
CInt16/CInt32/CFloat32/CFloat64}] [-strict]
[-if format]* [-of format]
[-b band]* [-mask band] [-expand {gray|rgb|rgba}]
[-outsize xsize[%]|0 ysize[%]|0] [-tr xres yres]
[-r {nearest,bilinear,cubic,cubicspline,lanczos,average,rms,mode}]
[-unscale] [-scale[_bn] [src_min src_max [dst_min dst_max]]]* [-exponent[_bn] exp_val]*
[-srcwin xoff yoff xsize ysize] [-epo] [-eco]
[-projwin ulx uly lrx lry] [-projwin_srs srs_def]
[-a_srs srs_def] [-a_coord_epoch <epoch>]
[-a_ullr ulx uly lrx lry] [-a_nodata value]
[-a_scale value] [-a_offset value]
[-nogcp] [-gcp pixel line easting northing [elevation]]*
|-colorinterp{_bn} {red|green|blue|alpha|gray|undefined}]
|-colorinterp {red|green|blue|alpha|gray|undefined},...]
[-mo "META-TAG=VALUE"]* [-q] [-sds]
[-co "NAME=VALUE"]* [-stats] [-norat] [-noxmp]
[-oo NAME=VALUE]*
src_dataset dst_dataset
If several outputs were supported the synopsis would show dst_dataset*
I do not see especially good reasons to add such feature. The other command options have often connection to outputformat so user should be able to give different options for each output format "for JPEG use compression use QUALITY=85, for PNG use ZLEVEL=8". I think that would be at least as complicated for users as to run the command twice. In most cases GDAL would still need to run the process two times so there would be no savings in processing time.
If the data source is slow or hard to process it could make sense to cache the data into some speedy format and use that for the subsequent processes. However, users can do that themselves relatively easy for example by reading data from a slow WFS service into a GeoPackage file and by using the local .gpkg file for next conversions
For more complicated ETL processes with several outputs I would have a look at dedicated tools like the Graphical modeler in QGIS.