Medicaid will only be primary if there is no other policy for the subscriber. If you have a medicaid policy and you move from an active insurance policy to a Cobra policy medicaid will still remain as secondary. Additionally, if you are on cobra policy and deemed eligible for medicaid the cobra policy will still be the primary policy.
Medicaid is always a payer of last resort. Cobra is insurance you can purchase in order to maintain coverage if you are no longer covered by your employer. Since it is derived from commercial coverage, Cobra is always primary over Medicaid.
Medicaid is always considered the secondary insurance, so the Cobra would be primary. Even if there are no benefits with the primary insurance. The office will still have to bill your Cobra plan to get the denial in order to be able to bill Medicaid and collect payment for the service.
Medicaid will never be listed as someone's primary insurance if the patient is covered by more than one plan. Most likely your Cobra plan will be terminated if you have another Insurance.