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minecraft:underwater_mount_breathing is an entity component: it goes in the components object of a custom mob, or in one of its component groups. Lets riders of this entity breathe underwater while they stay mounted, as with the nautilus.
Example
Inside the components object of BP/entities/<name>.json (values are the defaults or placeholders):
BP/entities/<name>.json
{
"minecraft:underwater_mount_breathing": {}
}