First thing to is run the mixerset utility that installed as part of the dos drivers, typically c:\sb16 and double check you have left and right gain settings matching.
Which games? have you tested. different games "talk" to the soundcard in different ways.
Finally if you play the same dos games from within Windows do you have the same problem?

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I tested it in Doom, Wolfenstein, The Adventures of Maddog Williams, Prince of Persia (intro music). Did not try playing them on Windows. Windows 95 games were fine. I used https://www.philscomputerlab.com/creative-drivers.html driver sbw9xup.exe. The same game works fine with ESS ISA 1868F or SB Live! 5.1.

I will check out that mixerset.

I have checked under the Windows 95 and those games also have mono music. Mixer settings are ok there. On one site I found this information:

In its early stages, Sound Blaster 16 retained the Yamaha OPL3 chip for FM music synthesis, so was still backward-compatible with the original Sound Blaster. Unfortunately, due to a flaw in the initial silicon, the Sound Blaster 16 range do not have proper Sound Blaster Pro compatibility (1 or 2) - so when it ran in "Sound Blaster Pro compatibility mode" (i.e. for all DOS games), it could only support mono playback.

Sorry for double post.

Seems that the music is playing through the OPL3 chip on the SB16 since you mentioned Wolfenstein 3-D. The music should normally play on both speakers if the output is mono. As mentioned before make sure that the left and right gain for the MIDI volume (the OPL3 FM synthesizer chip volume is tied to this one) are set in the middle on the mixerset utility.

This has nothing to do with the SB16 not being SBPro compatible, which is a totally different issue and only affects a specific stereo PCM playback mode in this case.

OK, had a chance to double check. I don't have a Vibra16 but I checked against my AWE32 which should act the same if setting it up as a SB16

I tested Doom, Wolf3d and Prince of Persia all play music in both speakers as I thought.
Doom is modern enough to have SB16 drivers so shouldn't have any compatibility issues anyway.

The SBPro compatibility issue is well known, even back in the day but as said above you loose the ability to hear different sounds in the left and right speakers (stereo) and just get the exact sounds coming from both (mono)

You cold try the line out rather then speaker out socket (or visa versa)

Duke3d, Decent, Warcraft 2 are my preferred games for dos sound troubleshooting as you can check the sound settings direct from the setup program.
Even the shareware versions are enough for testing