OK I just realized that I missed something important when I looked at that section: "and"!
(1) It retransmits only communications from GMRS stations operating under authority of the individual license under which it operates; and,
(2) The GMRS stations whose communications are retransmitted are properly identified in accordance with this section.
Notice that it also says "under which it operates", so it's talking about the repeater, not about the stations using the repeater. So if you stand up a repeater that is only going to be used by your family, under your GMRS license, and all the users identify as required, the repeater does not have to ID.
Otherwise, the repeater is required to ID every 15 minutes as described in that same section, although it is implied that this is only while the repeater is in use.