Apparently I have 623 backup files in that folder.
Are you sure of that? How many
folders show up at the top level of the folder 'Backup'? These will be folders with unique IDs for names (i.e. long hexadecimal strings). Each
folder is a backup. Inside each folder are many files (hundreds or thousands) that are all part of that backup.
iTunes only keeps one backup, unless you interrupt a backup (leaving a damaged one) or manually create one in iTunes (secondary-click device and select Backup). So, for example in my Backup folder there are three subfolders - one for each iPhone (two syncing to this computer) and one for an iPod Touch. This can be confirmed in iTunes > Preferences > Devices tab, which lists the three backups by device name and date (and the dates match the modified dates on the folders).
If you
really have 623 backups (i.e. you have 623 folders in the Backup folder and 623 backups show up in iTunes Preferences), you have something very odd occurring.