I was always under the impression that Linux 386 simply meant 32 bit support. So, no 386 support would mean no 32 bit support rather than the 386 specifically.
I have seen i386 being used in a couple of contexts. For example, Debian uses it to refer to the 32-bit port[1]. It also seemed to be used when various Linux distributions built Pentium optimized packages when that became a thing. In that case 386 would be used for earlier 32-bit processors and 686 would be used for later 32-bit processors. So the nomenclature is not always clear.
[1] https://www.debian.org/releases/stable/i386/