These color backgrounds provided in the choice columns are only for the viewing purpose. It will not come in exported excel/csv when user clicks on OOTB "Export to Excel" feature.
To achieve this, the only option left is thru customizations. There are several options available in the customizations as follows:
Power Automate Desktop: Using this, you can apply color backgrounds to the cell but it is a very tedious approach. First we have to fetch all data from the SharePoint list -> create excel file and add colors to the cells. It is very static because it requires one machine to do such operations. More details you can find here.
SPFx Listview CommandSet button: Using this, you can create one custom commandset button on your list view. You can write logic that upon the button click, it will fetch all data -> create excel file and add colors to the cell -> download it in the browser. You can use xlsx-js-style react library to create the excel file. You can use JavaScript/typescript code for the same.
Azure Web Job: You can write C# console application to fetch all list data and to prepare the excel file(along with colors) as per your need. You can save this excel file in one document library and can share its url to the urls. Since it is the Azure web job, you can schedule it to trigger once in a day, thrice in a day...anything as per your need.
xlsx-js-stylelibrary with react for the same.