Dynamically adding list

is there a way to create the list or rename the list for example from tempList to Collection1?

NO!

What would be the use case for this? The user doesn't write/read/care about the code ... you as a developer are ... after that your code is compiled and you don't just extend the compiled code afterwards on runtime ;)

It sounds like what you want to use is a Dictionary<string, List<int>> so you can address different Lists via a string key name like e.g.

private Dictionary<string, List<int>> _lists = new Dictionary<string, List<int>>();

// list is optional => if you already have a list you want to register pass it in
// if only a string is passed in a new list is automatically created
public void AddList(string keyName, List<int> list = null)
{
    if(_lists.ContainsKey(keyName))
    {
        Debug.LogError($"Already contains a list with key name {keyName}", this);
        return;
    }

    if(list == null) list = new List<int>();

    _lists.Add(keyName, list);
}

public List<int> GetList(string keyName)
{
    if(!_lists.ContainsKey(keyName))
    {
        Debug.LogError($"Does not contain a list with key name {keyName}", this);
        return null;
    }

    return _lists[keyName];
}
// or alternatively
public bool TryGetList(string keyName, out List<int> list)
{
    return _lists.TryGetValue(keyName);
}

public void RemoveList(string keyName)
{
    if(!_lists.ContainsKey(keyName))
    {
        Debug.LogError($"Does not contain a list with key name {keyName}", this);
        return null;
    }

    _lists.Remove(keyName);
}

Now IF by "User" you actually mean a developer using your script in Unity it can get a bit more complex but you could do something like

[Serializable]
public class ListEntry
{
    public string Name;
    public List<int> Values;
}

// Edit via the Inspector in Unity
public ListEntry[] lists;

then you can simply add and name your lists via the Inspector.

I would then still use the before code (or something similar) on runtime and simply create the dictionary once using the values from the Inspector e.g.

private void Awake()
{
    foreach(var list in lists)
    {
        _lists.Add(list.Name, list.Values);
    }
}

And of course you then can/should implement something that makes sure that all list names are unique. Either in OnValidate or in a custom editor script .. but I think that is a bit over the scope of this question ;)

Variable names, have no value in your code. They are something like placeholders of real values that are referenced around your code. Since there is no use case for what you ask, the answer is no.

If you want to create/access a list based on a key, then you could create a dictionary holding lists.

Dictionary<string,List<int>> lists = new Dictionary<string,List<int>>();

Then, when the new user enters

lists.Add("username", new List<int>);
// or if you want to use and existing
lists.Add("username", existingIntList);

And access like this:

// i.e.
foreach (int userItem in lists["username"]) {
  // etc
}