I know some are saying that they enjoy kinda giving people a light ribbing and making fun of them as 'banter'.
However. I'd say don't beat yourself up for finding it upsetting.
I grew up in a family where we did not make anyone the 'target' of jokes. Humour was usually absurd and silly but not the mocking, 'making-fun' kind of banter.
My flatmate, on the other hand, makes very biting 'funny' comments to other people (and me) about our choices, our feelings, our opinons, our (ahem) food choices... anything.
I found this nastiness (in an otherwise lovely person) kind of mistifying, until I saw her interacting with her siblings. They were brutal to each other. It was just constant mockery, sarcastic comments, 'funny' criticism of each other.
In an ideal world, each person should have the maturity and self-awareness to modify their behaviour if it's upsetting other people. To continue to mock and make fun of someone who is not enjoying it - well, the people doing that KNOW what they're doing. They're being dicks.
Really stupid, tiny example: last night, I was making myself some home-made soup when my flatmate came into the kitchen, looked at it and said 'eeew what the fuck is that?' - and laughed. I'm sure in her family that's probably the starting point for a hilarious ribbing. I just thought 'can she ever stop saying rude things' and ignored her. Queue a bad atmosphere between us. But you know what - I'm not her family. I'm not her sibling. I'm a completely different person. It is totally inappropriate to just start mocking someone if they are not the kind of person who can enjoy mockery. I honestly don't even find it funny. It's just dreary, depressing and rude. It's also unimaginative. There are so many OTHER, non hurtful things in the world to talk about, find humour in, get mega laffs out of.
I realise we all decide for ourselves how much we let something affect us; so I'd advise putting a protective-shell into action, much like a tortoise, if you are aware of this happening, but most importantly, don't turn it in on yourself and think 'I'm being overly sensitive'. Last night I just thought 'she's being a dick because her whole family talk like that all the time'. I tried to ignore it and not feel compelled to show that I was a 'good sport' or some other such BS.
In summary, I don't think there's anything wrong with you for feeling upset about it.