@robert-hh
What about:

ifneq ($(MICROPY_PY_CC3K),1)
GIT_SUBMODULES += lib/wiznet5k
endif

in Makefile after line 38:
GIT_SUBMODULES += lib/libhydrogen lib/lwip lib/mbedtls lib/stm32lib

There's already a conditional GIT_SUBMODULES += lib/wiznet5k in the block of ifneq ($(MICROPY_PY_NETWORK_WIZNET5K),0). And I placed that for tests already after the other GIT_SUBMODULES statement. at line 38. No change.
It's more a problem in the way CI builds the test environments, by not (always) including the required submodules. resp. not setting the proper flags when doing so. So it looks like it uses the same build environment for all Wiznet and CC3K variants.

There's already a conditional GIT_SUBMODULES += lib/wiznet5k in the block of ifneq ($(MICROPY_PY_NETWORK_WIZNET5K),0). And I placed that for tests already after the other GIT_SUBMODULES statement. at line 38. No change.

I propose to do a conditional GIT_SUBMODULES += lib/wiznet5k with a CC3K condition instead:

# CC3K vs WIZNET driver incompatibility workaround
ifneq ($(MICROPY_PY_CC3K),1)
GIT_SUBMODULES += lib/wiznet5k
endif

It might not be an obvious workaround if someone reads the makefile in the future, but I kind of remember that you told me that this CC3K was obsolete, no?

No. That does not help. It looks like the problem is in ci.sh. The respective section is below:

function ci_stm32_pyb_build {
    make ${MAKEOPTS} -C mpy-cross
    make ${MAKEOPTS} -C ports/stm32 submodules
    git submodule update --init lib/btstack
    git submodule update --init lib/mynewt-nimble
    make ${MAKEOPTS} -C ports/stm32 BOARD=PYBV11 MICROPY_PY_NETWORK_WIZNET5K=5200 MICROPY_PY_CC3K=1 USER_C_MODULES=../../examples/usercmodule
    make ${MAKEOPTS} -C ports/stm32 BOARD=PYBD_SF2
    make ${MAKEOPTS} -C ports/stm32 BOARD=PYBD_SF6 NANBOX=1 MICROPY_BLUETOOTH_NIMBLE=0 MICROPY_BLUETOOTH_BTSTACK=1
    make ${MAKEOPTS} -C ports/stm32/mboot BOARD=PYBV10 CFLAGS_EXTRA='-DMBOOT_FSLOAD=1 -DMBOOT_VFS_LFS2=1'
    make ${MAKEOPTS} -C ports/stm32/mboot BOARD=PYBD_SF6
    make ${MAKEOPTS} -C ports/stm32/mboot BOARD=STM32F769DISC CFLAGS_EXTRA='-DMBOOT_ADDRESS_SPACE_64BIT=1 -DMBOOT_SDCARD_ADDR=0x100000000ULL -DMBOOT_SDCARD_BYTE_SIZE=0x400000000ULL -DMBOOT_FSLOAD=1 -DMBOOT_VFS_FAT=1'

If I run the line
make ${MAKEOPTS} -C ports/stm32 BOARD=PYBV11 MICROPY_PY_NETWORK_WIZNET5K=5200 MICROPY_PY_CC3K=1 USER_C_MODULES=../../examples/usercmodule
locally, I get the same error messages. MICROPY_PY_NETWORK_WIZNET5K=5200 and MICROPY_PY_CC3K=1must not be combined in one compile.

I'll close that for now to do a silent test for ci.sh changes.

Apparently the main issues are:

drivers/cc3000/inc/socket.h:67: error: "SOCK_STREAM" redefined
drivers/cc3000/inc/socket.h:67: error: "SOCK_DGRAM" redefined

I will try to do a #ifdef #undef to see if it goes further

Edit: No chance. It's a matter of include path precedence... Impossible to solve without, as you said, prevent make call with multiple network drivers.

By the way, the MICROPY_PY_NETWORK_CYW43=1 does not seem to compile, at least on my machine for a missing definition of symbol pyb_pin_WL_SDIO_1 the compiler is proposing to use pyb_pin_WL_SD_D1 instead. Was eventually this pin renamed amongst the various versions?
Our work looks like the Danaïdes' barrel :-). Every time you fix a thing a new one blows up.

I have a proper set-up, ci passes. I'm just cleaning it now and run a further test.

I changed ci.sh in splitting the builds for wiznet5k and cc3k. But to be safe, I opened a new PR. I could not tell that after re-opening all commits were included.