userChildren = [{id: 'id', name: 'name', dateOfBirth: '2010-01-01'}]
await axios.patch(
        `${config.iamHost}/api/user/${session.userId as string}`,
        {
          user: {
            email: session.user!.email!,
            data: {
              children: userChildren,
            },
          },
        },
        {
          headers: {
            "X-FusionAuth-TenantId": config.iamTenantId,
            Authorization: config.iamApiKey,
          },
        }
      );

The code above did change the user custom data to

children.id id
children.name name
etc

I think because I didn't put the properties in the double quote as you did.

So I have found a workaround, which is parsing it as a JSON string and later parsing it back.

data: {
   children: JSON.stringify(userChildren),
},

Here's a data object which should work:

data: {
  children: [ {
    "name": "foo"
     "value": "bar"
  } ],
}

Can you please print out the JSON of the object you are trying to store in the user.data field and share it?