Details: https://forum.unity.com/threads/solution-layoutelement-fit-parent-with-aspect-ratio.542212/

LayoutElementFitParent.cs

using UnityEngine.UI;

[RequireComponent(typeof(RectTransform), typeof(LayoutElement))]
[ExecuteInEditMode]
public class LayoutElementFitParent : MonoBehaviour
{
	[SerializeField] private float aspectRatio = 1;

	[SerializeField] private bool updateMin = false;
	[SerializeField] private bool updatePreferred = false;

	private bool isDirty = false;
	private Vector2 lastParentSize;

	private new RectTransform transform;
	private LayoutElement layoutElement;

	public float AspectRatio
	{
		get { return aspectRatio; }
		set
		{
			aspectRatio = value;
			isDirty = true;
		}
	}

	public bool UpdateMin
	{
		get { return updateMin; }
		set
		{
			updateMin = value;
			isDirty = true;
		}
	}

	public bool UpdatePreferred
	{
		get { return updatePreferred; }
		set
		{
			updatePreferred = value;
			isDirty = true;
		}
	}

	private void OnEnable()
	{
		transform = GetComponent<RectTransform>();
		layoutElement = GetComponent<LayoutElement>();

		isDirty = true;
	}

	private void Update()
	{
		Vector2 parentSize = GetParentSize();

		// Mark as dirty if parent's size changes
		if (lastParentSize != parentSize)
		{
			lastParentSize = parentSize;
			isDirty = true;
		}

		// Only recalculate layout size if something has changed
		if (!isDirty) return;
		isDirty = false;

		float neededWidth = parentSize.y * aspectRatio;
		float neededHeight = parentSize.x / aspectRatio;

		// Is height the limiting factor?
		if (neededWidth <= parentSize.x)
		{
			// Scale to match parent's height
			SetSizes(neededWidth, parentSize.y);
		}
		else
		{
			// Scale to match parent's width
			SetSizes(parentSize.x, neededHeight);
		}
	}

#if UNITY_EDITOR
	private void OnValidate()
	{
		// Inspector fields have changed, mark as dirty
		isDirty = true;
	}
#endif

	private void SetSizes(float x, float y)
	{
		if (updateMin)
		{
			layoutElement.minWidth = x;
			layoutElement.minHeight = y;
		}
		if (updatePreferred)
		{
			layoutElement.preferredWidth = x;
			layoutElement.preferredHeight = y;
		}
	}

	private Vector2 GetParentSize()
	{
		var parent = transform.parent as RectTransform;
		return parent == null ? Vector2.zero : parent.rect.size;
	}
}