[e: i apparently mistook discordio for a couple of the other discovery platforms we utilized, but it seems to be the same concept as the others, looking at the web archives]
discordapp back then was really just a collection of servers pretty siloed/insulated from each other, with barebones voice and text chat functionality.
discordio offered some basic form of discovery and cross-server exploration/networking when it was effectively nonexistent back then. it, along with other outreach efforts on our part, certainly helped boost our community size and amusingly also attracted a lot of teens approaching us to ask if we wanted to "partner up" with their server (i help administer a studygroup server on discord).
anyway, 'partnered', to my recollection, means a formal arrangement with discord where the particular server community is directly promoted by discord on front pages and such, in exchange for meeting a higher bar of conduct that represents a model community (SFW, PC, etc.).
one of the perks that comes with this is being granted "Level 3" boost status, free of charge (normally costing anywhere from $49-70/mo, depending on circumstances), which is what directly grants the custom link feature.
Do you still have it? I believe a discord I was on had it too, but then they made it a paid feature (boosting) and started taking it away from smaller discords. Or maybe yours got grandfathered.