In short, you need to use IDs not labels to set the field. Long version with an example:
I have a multi-value field for a Person that can take one of three values
In order to set this field, first I need to call GET /personFields (docs) and find my custom field in the response to get ids
After that I can call PUT /persons/<personId> with the following body:
{
"2c804c080cb2fbba1c9d9c335ff3778dad8308af": "51, 52"
}
Where 2c804c080cb2fbba1c9d9c335ff3778dad8308af is a custom field key, and "51, 52" values I want to set.
As result, a person got these values
Hope it clarified things.