What you are asking for is a "server side copy". Yes rclone supports this as do s3, azureblob and gcs.
The main limitation is that you should use the same remote, so do
rclone copy s3:bucket1 s3:bucket2
Not
rclone copy s3:bucket1 anothers3:bucket2
If you run with -v you can see in the logs whether the copies are done with server side copies.