is it possible to move each color on its own layer?

Hi, here's a simple script that will arrange all pathItems of a specific color into separate layers:

function deliminateColors() {
    var doc = app.activeDocument, items = [];
    for (var i = doc.pathItems.length - 1; i >= 0; i--)
        doc.pathItems[i].filled ? items.push(doc.pathItems[i]) : null;
    items.forEach(function (item) {
        checkForLayer(rgbToHex(item.fillColor)) ? addToLayer(item) : createLayer(item);
    });
}
function checkForLayer(name) {
    var doc = app.activeDocument, found = false;
    for (var i = 0; i < doc.layers.length; i++)
        if (new RegExp("^" + name + "$").test(doc.layers[i].name))
            found = true;
    return found;
}
function createLayer(item) {
    var layer = app.activeDocument.layers.add();
    layer.name = rgbToHex(item.fillColor);
    layer.color = item.fillColor;
    addToLayer(item);
}
function addToLayer(item) {
    var layer = app.activeDocument.layers.getByName(rgbToHex(item.fillColor));
    item.move(layer, ElementPlacement.PLACEATEND);
}
function rgbToHex(color) {
    return "#" + [color.red, color.green, color.blue]
        .map(function (c) {
            c = c < 256 ? Math.abs(c).toString(16) : 0;
            return c.length < 2 ? "0" + c : c;
        }).join("");
}
Array.prototype.forEach = function (callback) {
    for (var i = 0; i < this.length; i++)
        callback(this[i], i, this);
};
Array.prototype.map = function (callback) {
    var mappedParam = [];
    for (var i = 0; i < this.length; i++)
        mappedParam.push(callback(this[i], i, this));
    return mappedParam;
};
deliminateColors();

Results in Layers panel on the left with a 16-color image trace for simplicity:

Since you're using Image Trace and didn't give too much detail, I assumed you only wanted fills to be checked, item names intact, layer names to be the specified color in hexadecimal form, the layer label to be equal to this color (though this would make selection hard, comment out the layer.color line in createLayer() to remove), and didn't touch the original layers just in case there's more to the document than these items.

Actions will get you a long way.

You can start by opening an image in Illustrator, this has the advantage that the Swatches panel is empty.

Unfortunately Image Trace cannot be recorded with a preset, so you need to select one manually.

The rest can be semi automated:

An action to expand the Image Trace and adding the colors to the Swatches Panel

When a swatch is selected, an action can select the same fill colors, cut the selection, create a new Layer (recorded by clicking the New Layer Icon with the Cmd (or Ctrl for Win) key depressed to avoid recording naming the layer) and paste the result. You need to play this action for every color.

When you say you are "doing it manually", you don't mean one object at a time--do you? Are you using the Select>Same>Fill Color menu? If you combine that with the Collect in New Layer menu option, you should be able to make an Action.

yes it is possible. but it will also depend on your artworks appearance layers. scripting has no way of cycling through appearance fills, and strokes to determine what is the top fill. it will also get tricky to determine the fill color of a group vs the fill color of each group member if you have any.