Solution: CounterContext

Here is a simple solution:
class CounterContext :
    AbstractCoroutineContextElement(CounterContext){

    private var counter = 0

    fun next(): Int = counter++

    companion object : CoroutineContext.Key<CounterContext>
}
Here is a thread-safe solution:
class CounterContext :
    AbstractCoroutineContextElement(CounterContext){

    private val counter = AtomicInteger(0)

    fun next(): Int = counter.getAndIncrement()

    companion object : CoroutineContext.Key<CounterContext>
}

Example solution in playground

import kotlinx.coroutines.*
import org.junit.Test
import java.util.concurrent.atomic.AtomicInteger
import kotlin.coroutines.*
import kotlin.test.assertEquals

class CounterContext :
    AbstractCoroutineContextElement(CounterContext){

    private var counter = 0

    fun next(): Int = counter++

    companion object : CoroutineContext.Key<CounterContext>
}

fun main(): Unit = runBlocking(CounterContext()) {
    println(coroutineContext[CounterContext]?.next()) // 0
    println(coroutineContext[CounterContext]?.next()) // 1
    launch {
        println(coroutineContext[CounterContext]?.next())// 2
        println(coroutineContext[CounterContext]?.next())// 3
    }
    launch(CounterContext()) {
        println(coroutineContext[CounterContext]?.next())// 0
        println(coroutineContext[CounterContext]?.next())// 1
    }
}

class CounterContextTests {
    @Test
    fun `should return next numbers in the same coroutine`() = runBlocking<Unit>(CounterContext()) {
        assertEquals(0, coroutineContext[CounterContext]?.next())
        assertEquals(1, coroutineContext[CounterContext]?.next())
        assertEquals(2, coroutineContext[CounterContext]?.next())
        assertEquals(3, coroutineContext[CounterContext]?.next())
        assertEquals(4, coroutineContext[CounterContext]?.next())
    }

    @Test
    fun `should have independent counter for each instance`() = runBlocking<Unit> {
        launch(CounterContext()) {
            assertEquals(0, coroutineContext[CounterContext]?.next())
            assertEquals(1, coroutineContext[CounterContext]?.next())
            assertEquals(2, coroutineContext[CounterContext]?.next())
        }
        launch(CounterContext()) {
            assertEquals(0, coroutineContext[CounterContext]?.next())
            assertEquals(1, coroutineContext[CounterContext]?.next())
            assertEquals(2, coroutineContext[CounterContext]?.next())
        }
    }

    @Test
    fun `should propagate to the child`() = runBlocking<Unit>(CounterContext()) {
        assertEquals(0, coroutineContext[CounterContext]?.next())
        launch {
            assertEquals(1, coroutineContext[CounterContext]?.next())
            launch {
                assertEquals(2, coroutineContext[CounterContext]?.next())
            }
            launch(CounterContext()) {
                assertEquals(0, coroutineContext[CounterContext]?.next())
                assertEquals(1, coroutineContext[CounterContext]?.next())
                launch {
                    assertEquals(2, coroutineContext[CounterContext]?.next())
                }
            }
        }
    }
}