*Khem, khem*
The idea for this question came to me not so long ago, but it really interested me, thank you for your help in formulating the question, with your permission I will use your formulation;)
What tickling is for each mercenary?
Again, you're welcome. Now, without further ado
Scout: For him it's mostly an extremely pleasant physical sensation, so pleasant but also so unusual he's embarassed by it. Yes, it's also great for making him feel happy and cozy and of course that guy craves human interaction but despite that he's not that good at it. But but if someone tickles him it pretty much has to mean they're close, right? Having said all that this all stems from the whole pleasant physical sensation first and foremost.
Soldier: It's a way to discipline, either others or himself as well as a way to challenge his strenght and willpower since he's all about that really. And you know, he's insane too like that. He does genuinely enjoy the tickling in itself either as a lee or a ler but this sentiment is buried deep in his mind under all the lead poisoning, letting itself known to him ocasionally.
Pyro: Wooo boy what tickles aren't for Pyro? They're a way to bond, an activity to pass time, way to make more friends and spread affection but most importantly - they're a way to have fun. Making others laugh and squirm and make all tose funny noises and doing all that himself is just pure fun for Pyro.
Demoman: It's a way to get his teammates to loosen up and have fun with him plus it's a solution to so many things. Sniper's being grumpy and does not want to leave his camper van? Tickles. Engie's being annoyed that his projects aren't working out? Tickles. Medic getting a bit too crazy for comfort? Tickles. It helps with tension, to keep it short.
Heavy: It's a fun way to utilize his strenght without hurting anyone for once (as ler) as well as something that causes comfort (as both ler and lee). He already loves doing basically both of those things to the people he cares about and, you know, is not trying to actively kill, but then add tickles to those and it's even more fun.
Engineer: He himself is not fully aware of that, but tickling is a sort of replacement of the comfort and good time he lacked when he was younger due to his family's legacy (I know you said you didn't read the comics so I'll just say the whole fmaily legacy ting isn't something I made up on the spot, this is canon). It's just straight up fun and relaxing, but he's embarassed that something silly like that makes him feel so good.
Medic: It's a way to satisfy his sadistic urges without actually spilling any blood etc, this goes for him as a ler but as a lee too. How? Well, it's a way of testing this "sadistic" method on himself to know fully exactly what it feels like. That's how it started anyway because eventually he realized just how much he loves the tickling not just because it's sadistic in any way but also because it's just straight up fun.
Sniper: It's a way for him to open up and keep relationships more easily, believe it or not. He likes his alone time, don't get me wrong, but sometimes he remembers he actually craves positive interactions with others. Sometimes. And being tickled/tickling someone else is sure one if weird way to do exactly that. Listen, I can absolutely see him one day just sitting up in his bed, thinking "Crikey, I need to hang out with someone n o w," and then dragging someone Scout to his van to tickle the heck out of them.
Spy: It's a unigue form of intimacy, romantic or otherwise and dang he likes intimacy, just in general. And yes, a form that is extremely embarassing to him but one he can't help but love. As a ler it's an extremely good way to prove his superiority exactly because he knows how embarassing it is to most, let alone freaking mercenaries so when he can take them down like that it's just amusing.