Just from a geographical perspective this seems like an odd pairing. Never played the game and probably never will.
They're just code names, and the characters are from the same city.
Partially correct, as "Texas" is her family name - it's repeatedly alluded to in her bio. She's hardly the only Operator to use her real name as her codename either, probably helps that like most such she has a technically perfectly legal full-time job.
"Lappland" conversely is clearly a codename; she's a former(?) Mafia-analog assassin from the local Not Italy region ("Siracusa") and folks from there appear to have appropriately themed names like Angelina or the Super Mafia Bros Capone and Gambino in the recent event storyline (who, uh, Lapp is heavily implied to have killed, gratuitously and p much for shits & giggles, in her epilogue cameo appearance...).
Moreover Texas is Columbian by birth (apparently the generic Western Europe/North America equivalent country, as distinct from eg. the British-themed Victoria, Totally Not Imperial Russia Ursus or Not Poland Kazimierz) but is very heavily hinted to have quite the dark past in the Siracusan underworld - the mobster Gambino readily recognised the name and not in a good way - which is where her connection with her near-stalker Lappland apparently stems from. Lapp keeps complaining Texas has "gone soft" as she doesn't murder the shit out of every mofo who crosses her path anymore these days... though she's happy to pick up the slack herself as the two late and unfortunate Mafioso brothers could posthumously attest.
Uhh..
Lappland suffers from an illness/infection that will kill eventually her, and it causes various damage to the body over time. So Lappland making a joke about it caused Texas to be upset, because it's a serious matter. A lot of major characters in Arknights have it, and they're still searching for a cure.
The illness - properly Oripathy (after the magical mineral Originium you catch it from) and colloquially just Infection - or rather the pariah status of most of those afflicted with it is indeed the background driving force of the major conflict in the storyline, with no few hints of even darker undertones that border on lowtier Cosmic Horror...
Good for her Lapp's present employer is at the bleeding edge of medical research on it (she isn't even their worst-Infected patient), and while even they still can't cure it they can largely halt its progress which is sorta important because the disease is invariably fatal untreated.
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