Mighty interesting! There is a clue in the FCOM:
Four frequencies per FMGC
If you check FCOM Chapter 22 § 20-20-30 you'll note it says:
Each FMGC [Flight Management and Guidance Computer] automatically uses its four DME frequencies as follows:
One DME frequency for display. It is possible to tune it manually or automatically. This DME frequency is also used for VOR/DME position
Two DME frequencies in DME/DME mode for calculating the radio position of the aircraft. The FMGC automatically tunes these as a
function of their best accuracy. The flight crew does not receive any
indication that this process is happening.
One DME frequency is connected to the ILS/DME.
Note my bold emphasis above. Similarly for the A380, in clearer language:
In normal operation, each FMS can tune four DME s at the same time. (...)
Scanning DME (distance measuring equipment) that check the distances from five different DME stations simultaneously in order to determine one position every 10 seconds [emphasis mine]
—Wikipedia: Flight management system § Position determination
The patent for this scanning-type DME was filed in 1982:
(...) a scanning type DME unit which is rapidly sequenced to interrogate said multiple ground stations successively and determine distance thereto for all stations within range, and a computer to validate and process these distance measurements in sequence to provide an accurate computation and display of aircraft position and wind information. [emphasis mine]
—Meyer, Donald H. "Accurate DME-based airborne navigation system." U.S. Patent No. 4,583,177. 15 Apr. 1986.
Regarding the failure scenario, each FMGC can tune both its interrogator, and the other interrogator if the other FMGC failed. Each FMGC is still limited to four frequencies though. So, if one DME fails (antenna or its interrogator), you'll have two frequencies which you can control (one for display, and one for ILS), and two the FMGC will use for DME/DME position, but you can't have two different DMEs for display.
1: Airbus Single Aisle Training Manual ATA 34