C# Inheritance - Reference the INSTANCE of a variable, is it possible?

I’ve been digging around and have found many similar but not exactly matching results to this question, all of which have made this infinitely more confusing. Below is example mock-up code of what we’re trying to do here. Please read the bottom paragraph for an explanation. Thank you!

Base.cs

using UnityEngine;
using System.Collections;

public class Base : MonoBehaviour {

	public GameObject Chase;

	void Update () {
	Chase = GameObject.Find("Chase"); // Never GO.Find on Update, this is just a mock-up.
	}
}

Race.cs

using UnityEngine;
using System.Collections;

public class Race : Base {

	void Update () {
		Debug.Log(Chase);
	}

}

With that in-mind, is it possible to get the instance of a variable that we’re inheriting from? The idea here is having a ‘base’ class that houses a couple variables that will be used across all scripts within the project. These variables are assigned during run-time, so we need to grab the Instance of them. However at this point, I am no longer sure if this is something inheritance will support. Statics don’t seem like the way to go, and namespaces/interfaces are just as useless in this case scenario.

Singleton solution could look like this for example:

//add this class to your Chase gameObject and make sure it's in the Scene somewhere:
public class ChaseClass : MonoBehaviour {
	
	private static ChaseClass _singleton;
	public static ChaseClass singleton {
		get {
			return _singleton;
		}
	}

	public static GameObject go {
		get {
			return _singleton.gameObject;
		}
	}

	void Awake () {
		//address singleton
		if( _singleton==null ) {
			_singleton = this;
		}
		else if( _singleton!=this ) {
			Destroy( this.gameObject );
		}
	}

}

//this is how you will access this Chase gameobject from any class:
public class AnyClass : MonoBehaviour {

	void Update () {
		Debug.Log( ChaseClass.go );
	}

}