The origin of shoganai （しょうがない） comes from "shiyo ga nai" （仕様が無い） that generally means "be helpless", "be useless to", "(there's) no other choice", etc.
The expression "shoganai" or "shiyo ga nai" has two kinds of usage.
i) An idiom that means "be useless to", etc. as mentioned above.
In this context, "shoganai" is similar to "shikata ga nai" or "yamu o enai" (formal).
ii) An adjective that means "unruly" as in "shoganai kyaku" (an unruly customer).
Note that "shikata ga nai" and "yamu o enai" cannot be used in this context.
A formal analogue in this context is "te ni oenai" （手に負えない）.