The connection for benzos is somewhat tenuous in its own right, but gabapentin actually has basically nothing to do with Benzos and does not act on your GABA system
It is also not very potent but there is another drug that shares its mechanism that is potent in single digit mg amounts: pregabalin (Lyrica). It is potent and kinda sorta abusable (not really) and they both can produce a super mild but benzo-ish sedation for a couple hours for a day or two.
It is apparently a coincidence that the drug seemed to resemble GABA and got its name though. Strange.