You are POSTing from the client:
await client.PostAsJsonAsync("api/products", product);
not PUTing.
Your Web API method accepts only PUT requests.
So:
await client.PutAsJsonAsync("api/products", product);
You are POSTing from the client:
await client.PostAsJsonAsync("api/products", product);
not PUTing.
Your Web API method accepts only PUT requests.
So:
await client.PutAsJsonAsync("api/products", product);