Animals are sentient (like humans) so we should treat them like humans and not exploit (eat) them.

Vegans use this argument to promote veganism. Since we are not obligate carnivores, we should therefore not eat meat, which causes unnecessary harm. Since it's immoral to eat people, its immoral to eat animals.

That isn't even factually correct. Sentience just means being able to think. What they're actually looking for is sapience , which is a level of consciousness needed to qualify something as being a person equivalent to humans. You'd need to argue the personhood of a specific animal species in order to correctly declare it immoral to raise them for food, let alone keeping them in captivity due to becoming subject to slavery laws. Only a few species have passed the mirror test, which is used to determine their ability to recognize themselves as individuals. While the personhood of some of them is being debated, humans are still the only species currently recognized as people in the legal sense.

The code of ethics regarding sentient but non-sapient animals simply isn't the same as the one dealing with legal persons, though there are still regulations on the ethical treatment of livestock. Farms raising animals for meat in an ethical manner ensure they live comfortable and low stress lives before the slaughterhouse, so the real issue is in ensuring those standards are upheld and boycotting the brands you know don't do that.

Let's put this in logical form:

P1. Animals are sentient like humans
P2. We don't exploit humans by eating them
C. Therefore, we shouldn't exploit animals by eating them

Is it valid? No. Even if we grant both premises, it doesn't warrant the conclusion. Just because two things are alike in one way, doesn't make them equal. But what if we changed it up a bit...

P1. Animals are sentient beings.
P2. We shouldn't eat sentient beings.
C. Therefore, we shouldn't eat animals.

Now we have a valid argument in that, if we grant both premises we have to accept the conclusion. But this argument can never be objectively sound because we have a highly subjective premise (p2) and a (currently) non-falsifiable premise (p1). People's difference systems of morality will dictate if they agree with premise #2 or not.

This argument is loaded with several value claims—claims that are neither right or wrong but opinion or values. If I were arguing on behalf of the vegan, I might rephrase this argument using conditionals:

P1. Science overwhelmingly supports the idea that many non-human animals (including the ones we eat) are sentient.
P2. If you agree that sentience is the measure by which we should grant beings certain rights including the right not to be slaughtered for food, then
C. You must agree that we shouldn't eat animals.

This is a more convincing and reasonable argument, IMO. The major difference is, were not trying to trap people with bad logic; rather, we are allowing them the option of disagreeing with our condition that warrants the conclusion.