private val items: MutableList<T?> = mutableListOf() can be written more succinctly as private val items = mutableListOf<T?>() (the type is inferred and then no longer repeated (mostly) in an explicit type definition and the factory method call).

size can be backed by items.size:

 val size: Int
     get() = items.size

 fun push(item: T) {
     set(size + 1, item)
     siftUp(size - 1)
 }

 ...

 fun pop(): T? {
     ...
     set(0, get(size - 1))
     items.removeAt(items.lastIndex) // Remove original reference to the last item
     ...
 }

siftDown throws an AssertionError assertion when size == 0.

private val items: MutableList<T?> = mutableListOf() can be written more succinctly as private val items = mutableListOf<T?>() (the type is inferred and then no longer repeated (mostly) in an explicit type definition and the factory method call).

size can be backed by items.size:

 val size: Int
     get() = items.size

 fun push(item: T) {
     set(size + 1, item)
     siftUp(size - 1)
 }

 ...

 fun pop(): T? {
     ...
     set(0, get(size - 1))
     items.removeAt(items.lastIndex) // Remove original reference to the last item
     ...
 }

siftDown throws an AssertionError assertion when size == 0.

push and pop have different meanings in the Java Collections Framework associated with using a java.util.Deque as a stack. You might instead use the same verbs as java.util.PriorityQueue (namely add and remove) which can be used as a max heap:

 fun <T : Comparable<T>> maxHeap() = PriorityQueue<T>(Comparator.reverseOrder<T>())

private val items: MutableList<T?> = mutableListOf() can be written more succinctly as private val items = mutableListOf<T?>() (the type is inferred and then no longer repeated (mostly) in an explicit type definition and the factory method call).

size can be backed by items.size:

 val size: Int
     get() = items.size

 fun push(item: T) {
     set(size + 1, item)
     siftUp(size - 1)
 }

 ...

 fun pop(): T? {
     ...
     set(0, get(size - 1))
     items.removeAt(items.lastIndex) // Remove original reference to the last item
     ...
 }

The siftDown assertion on i should be assert(i >= 0 && i <= size) and not assert(i >= 0 && i < size).

private val items: MutableList<T?> = mutableListOf() can be written more succinctly as private val items = mutableListOf<T?>() (the type is inferred and then no longer repeated (mostly) in an explicit type definition and the factory method call).

size can be backed by items.size:

 val size: Int
     get() = items.size

 fun push(item: T) {
     set(size + 1, item)
     siftUp(size - 1)
 }

 ...

 fun pop(): T? {
     ...
     set(0, get(size - 1))
     items.removeAt(items.lastIndex) // Remove original reference to the last item
     ...
 }

siftDown throws an AssertionError assertion when size == 0.