I agree that it isn't a slur, it's a medical term. It can be used in offensive ways if one wants to though, just like "black" isn't an offensive term but calling people "you blacks" is.

Many people are ridiculed and made fun of for being fat. Fat people in movies are often used as simply comic relief and the whole thing about them is that they're fat. Take me, for example. I've never been obese. Growing up, I was called obese and made fun of for being chubby, while I was in a healthy weight range, just on the heavier side of it. I don't know if it's a cultural thing (my country viewing anyone who's not a BMI of 19 as fat) or a girl thing (I'm sure you all know girls who think they're fat and are really completely fine, this doesn't just come from nothing), but it was very deliberately being used purely as insult and mockery. Growing up, I IDENTIFIED as a fat girl. I thought my weight defined me as a person, I thought that was just who I was and there was nothing I could really do about it. (I'm not exaggerating I never really thought I was physically able to lose weight. I thought I was just worthless and a glutton and disgusting and that was just who I was and I didn't deserve anything good in life). This was due to me being an idiot obviously, but also how fat people are always used as a laughing stock rather than completely worthy and normal people who can just lose weight if they want to. Fat characters in movies are defined by their weight, very rarely do you see significant weight loss or gain in movies or TV shows, and when you do they make it look like they spent 100% of their time working out and went through this life-changing journey, which I obviously couldn't afford to do since I spent a lot of time sitting down studying. Through all this, again, I was never actually obese and was actually very active, but it is very easy for me to eat a little too much being a petite girl. So many eating disorders stem from bullying and harassment and obsessing over someone's weight. So many people overeat as a form of self harm or gain weight due to depression and not being able to get their ass off the bed. So many obese people suffer from mental illness. I'm very happy that you're not one of these and can have a healthy mindset about your weight. This is not always the case though and there are definitely people using fat as an insult and making fun of people because hehe jiggly bits are funny. Most fat girls I've talked to were terrified of working out and going running because they were constantly worried they looked ridiculous, that people would make fun of them for sweating too much, for being the fat girl at the gym, ... Offensive jokes are a different topic, but I personally do think certain words can be used with the sole intent of hurting people. It's not just about the literal word, but about what people's intentions with it are.

Again I'm not defending the girl on Twitter here, she does seem to be saying "obese" is universally a slur word, which I'm absolutely not. I do think there are nuances though.