You can work around this in this way:

using System.Collections.Generic;
using System.Reflection;
using UnityEngine;

public class DrawLineWithTexture:MonoBehaviour
{
    public Texture2D lineTexture;
    private OnlineMapsMarker marker1;
    private OnlineMapsMarker marker2;
    private OnlineMapsDrawingLine line;

    private void Start()
    {
        OnlineMapsControlBase.instance.OnMapClick += OnMapClick;
    }

    private void OnMapClick()
    {
        OnlineMapsMarker marker = OnlineMaps.instance.AddMarker(OnlineMapsControlBase.instance.GetCoords());

        if (marker1 == null)
        {
            marker1 = marker;
            return;
        }

        marker2 = marker;
        List<Vector2> points = new List<Vector2>
        {
            marker1.position,
            marker2.position
        };
        line = new OnlineMapsDrawingLine(points, Color.red, 10);
        line.texture = lineTexture;
        OnlineMaps.instance.AddDrawingElement(line);
        OnlineMaps.instance.OnMapUpdated += OnMapUpdated; // Delay because the line is not yet drawn.

        marker1.OnDrag += OnMarkerDrag;
        marker2.OnDrag += OnMarkerDrag;
        OnlineMapsControlBase.instance.OnMapClick -= OnMapClick;
    }

    private void OnMapUpdated()
    {
        UpdateUVScale();
        OnlineMaps.instance.OnMapUpdated -= OnMapUpdated;
    }

    private void OnMarkerDrag(OnlineMapsMarkerBase marker)
    {
        List<Vector2> points = new List<Vector2>
        {
            marker1.position,
            marker2.position
        };
        line.points = points;

        UpdateUVScale();

        OnlineMaps.instance.Redraw();
    }

    private void UpdateUVScale()
    {
        FieldInfo field = line.GetType().GetField("gameObject", BindingFlags.NonPublic | BindingFlags.Instance);
        if (field == null)
        {
            Debug.Log("No field");
            return;
        }
        GameObject lineGO = field.GetValue(line) as GameObject;

        MeshFilter meshFilter = lineGO.GetComponent<MeshFilter>();
        Vector3[] vertices = meshFilter.sharedMesh.vertices;

        float distance = 0;

        for (int i = 2; i < vertices.Length; i += 2) distance += (vertices[i - 2] - vertices[i]).magnitude;

        distance /= line.width * 2;

        MeshRenderer meshRenderer = lineGO.GetComponent<MeshRenderer>();
        meshRenderer.sharedMaterials[0].mainTextureScale = new Vector2(1, distance);
    }
}

Yes, this is a bad practice and a dirty hack, but I do not know how to do this without access to the vertices.
Theoretically, we can make gameObject field public.