You mentioned Frisk being tactile - so they probably don't mind at all if the monsters take every opportunity to pick them up, carry them around, hug them or hold them (and one time pretend to suplex them) -- Gallifreya
The first thing people noticed about Frisk was their silence. The second thing they noticed, assuming they got past the silence, was that Frisk was rarely out of the arms of a monster.
It didn't matter which one. Whatever came along to be Frisk's translator, that thing held Frisk as comfortably as they could manage. After a few days, there was even a kind of padded apron for the... less squishy monsters to wear.
The humans provided a translator, too. Just in case the monsters were interpreting Frisk to say whatever they wanted. They weren't. In fact, the monsters were more at home with Frisk's half-pantomime dialect than most translators were.
And for the home visits, Frisk rarely left the arms of their adopted mother. The giant goat creature with fangs.
"You don't have to hold them all the time," said the counsellor. "Frisk. I won't take you away. You're free to wander where you like."
"You need not fret," said Toriel. "Frisk is always free to go where they will." To demonstrate, she opened her arms.
Frisk leaned out and reeled an arm back in. Snugging it tight against them before resuming their pattern of patting.
"Frisk adores being hugged," said Toriel simply. "Having become familiar with their file... I would hazard the guess that they are making up for lost time."
Deep in Toriel's arm, Frisk nodded.
The front door slammed open. "What up, nerds!"
Frisk sat bolt upright. "Undyne!"
Toriel offered them egress. "Of course you can go play."
The counsellor followed Frisk to the front hallway, where a tall fish and a small dinosaur had their arms open to greet Frisk. Who ran into the arms of the fish, first. The counsellor almost had a heart attack when the fish appeared to suplex Frisk. But it was mock-wrestling. No danger to anyone.
Frisk giggled and immediately ran to the dinosaur, who simply picked Frisk up for a hug.
"You're still a weak little loser," grinned the fish. "But you're the best weakling I ever lost a battle to." More wrestling and some mock noogies, all of which were greeted with giggles, squeals, and Frisk tickling in return.
"Aaaah! Tiny hands! My only weakness. Noooo... I am defeated." Undyne overacted her way into a very theatric mock 'death'.
My God, it's full of nerds...
"Look what I brought," said the dinosaur. Alphys. She had a thick rectangle, "It's the box set for Mew Mew Kissy Cutie, the series!"
The counsellor left before the squealing got intense.