元组比较是按多个标准对元素进行排序的好方法，但是您弄错了元组元素的顺序。
元组以“字典顺序”进行比较，从第一个（最左边）元素开始。为了首先按名称排序，$0.name和$1.name必须是第一个元组元素。
因此，首先按名称（升序）排序，然后按计数（降序）排序，排序函数应为

array.sorted(by: {
    ($0.name, $1.count) < ($1.name, $0.count)
})

要基于多个条件进行排序，只需包含基于优先级相互级联的条件。
编辑：感谢Martin R指出错误，以下是修改后的代码：

array.sorted(by: {
    //First layer of filter. Sorts based on alphabetic order
    if $0.name < $1.name {
        return true
    }
    else if $0.name > $1.name {
        return false
    }
    //Second layer. Sorts based on greater count
    return $0.count > $1.count
})

输入：

let array = [
    Variant(name: "Bd", count: 55),
    Variant(name: "Bc", count: 55),
    Variant(name: "Ac", count: 12),
    Variant(name: "Ab", count: 12),
    Variant(name: "Ad", count: 8),
    ]

排序后：

Variant(name: "Ab", count: 12),
Variant(name: "Ac", count: 12),
Variant(name: "Ad", count: 8),
Variant(name: "Bc", count: 55), 
Variant(name: "Bd", count: 55)