How to assign custom colors to groups in a dispersion chart in Numbers

As far as I know there is no way to make the desired changes all at once, but there is a way that you can make things significantly faster than creating a brand new chart.

As it appears you've already done, copy and paste the chart into your Numbers worksheet. Depending on the chart you've created, select an individual line, bar, pie segment etc. and open the chart pane in the inspector. Here's a screenshot of what that looks like (with "Example 2" being the copied and pasted chart):

As you can see, in the field labeled "Data," Table 1 is referenced. This source can easily be changed to a different table. Just delete the listed source and select the new source of data from your worksheet (you don't have to type in the source). Depending on your chart you may also want to change the data source for the x-axis labels. This is also controlled by the inspector. That can be changed here:

While you'll have to change the data source for each line of the chart, it's certainly faster than creating a whole new chart and having to format it the way you'd like.

It appears that the Data Symbols series preference is not a part of the styles. Why can't you select all of the charts when you are finished with them (by Command clicking or Command + Shift dragging) and then changing the data point styling of all of them at one time?