I think that Cambridge definition is a little too specific. I'm more inclined to agree with the Oxford Dictionary:
commiseration (n): Sympathy and sorrow for the misfortunes of others; compassion.
To commiserate is to "share sadness" with someone. Although the noun form is not common, it can be used in that context.
There's no shame in relying on family members and close friends for support, love, and commiseration.
"Poor man!" thought Meredith in a sudden burst of commiseration.
As with any other saddening event, it can be used to offer sympathy on the loss of a competition:
Although outwardly happy, in private the friends of the silver-medal winner gave her their commiseration.
After their embarrassing defeat in the final round of the World Cup, some fans offered the national football team some commiseration, but many others suggested public execution.
Commiserations is also acceptable, as a way to offer sympathy, similar to how we might use congratulations:
Our commiserations on the loss of your father.
Note: I expect it's more common to use the verb commiserate than the noun commiseration. Also, condolences is significantly more common than commiserations.