20 isn’t actually that much encumbrance by itself. The “Fits” means that it fits you perfectly, so the encumbrance of the item is as low as it’s going to get.
It’s very possible that your buddy is a lot more skilled than you in melee combat, which you can check by talking to him and asking him to tell you about himself.
How much encumbrance you can handle depends mostly on how good your melee skills are. A character with no melee skill at all will struggle to hit things with 30 encumbrance, but someone with 10 melee skill and 10 cutting/bashing/piercing skill can wear 50 or more without a huge amount of trouble (though even then there are reasons to keep it low).
As for keeping it low, your only options are to find less encumbering clothing that does the job well enough, put up with it, or, if the clothing doesn’t already fit, you can refit it with a sewing tool if your tailoring is good enough, which can help a little.
It’s always a tradeoff, and you have to try and find the right balance. 20 encumbrance shouldn’t be too hard to deal with, the big reason you can’t hit the dog is because it’s considered a small creature, which makes it much more difficult to hit than, say, a regular zombie. You’ll get better at it as your skills increase.
