It's probably a bit more complicated than that. Cortical areas tend to be highly connected with their contralateral counterparts, so it's not really fair to expect that the effects of stimulation on one side of the brain doesn't impact the contralateral side.
The claustrum is itself a small gray matter region, but it is surrounded by white matter as well. Koubeissi et al used a fairly strong stimulus when they saw effects on consciousness; I have a hard time believing this stimulation was only affecting the claustrum and not fibers of passage at that stimulation intensity.
A recent study of 5 epilepsy patients attempted to replicate the Koubeissi finding, stimulating the claustrum (bilaterally in 4/5) in epilepsy patients, but none of the patients lost consciousness despite experiencing other effects. The authors suggest that the results presented in the earlier case study could be based on the unique circumstances of that patient, which include their epilepsy history and prior resection, and that the stimulation location in that Koubeissi case was not really directly in the claustrum. Additionally, Bickel & Parvizi were unable to use a stimulus as strong as the prior case report because their patients experienced negative side effects like painful muscle responses that Koubeissi did not see.
Bickel, S., & Parvizi, J. (2019). Electrical stimulation of the human claustrum. Epilepsy & Behavior.
Damasio et al, 2013 is another case study, where the patient had bilateral damage to a substantial portion of the brain, including the claustrum bilaterally. This patient had quite a bit of impact to various functions due to this damage, but they were still conscious. Therefore the answer to "Are both claustrums necessary for a conciousness" seems to be "no"; in a healthy brain it's a bit harder to say, because it's not ethical to damage the claustrum in healthy brains.
Damasio, A., Damasio, H. & Tranel, D. Persistence of feelings and sentience after bilateral damage of the insula. Cereb. Cortex 23, 833–846 (2013).
There are also other papers showing suppression or recovery of consciousness through stimulation of other brain regions besides the claustrum.
Redinbaugh, M. J., Phillips, J. M., Kambi, N. A., Mohanta, S., Andryk, S., Dooley, G. L., ... & Saalmann, Y. B. (2019). Central thalamus modulates consciousness by controlling layer-specific cortical interactions. bioRxiv, 776591.
Schiff, N. D., Giacino, J. T., Kalmar, K., Victor, J. D., Baker, K., Gerber, M., ... & Farris, S. (2007). Behavioural improvements with thalamic stimulation after severe traumatic brain injury. Nature, 448(7153), 600-603.