Many animal words in English have both positive and negative connotations, simultaneously. 'Dog' isn't always negative, but also also not always positive. The use of animal terms in relation of people is generally (but not always) used to denote a type of personality, not necessarily "good" or "bad". There are some exclusively bad or good though, so I'll try to make note of how each is perceived.
Swans are often used to describe grace and beauty, and are considered a superior animal compared to the duck. More upper-crusted in use. You can say something like, "She's an ugly duckling now, but she'll be a swan later." You can also say things like "She's a swan on the ice."... Downside is a swan is considered fragile and not down-to-earth. May sometimes be considered self-centered.
Calling something an "Ugly Duckling" saying they're ugly now, that they stand out or look worse than their peers now, but will be a swan later.
Which means young and vibrant. So you can say, "She's a real spring chicken" or "I'm not a spring chicken anymore." That said, this is kind of dated terminology, and will make you sound like you're not a spring chicken. Also rural in origin. Has the negative connotation of being unexperienced.
This means you're strong and independent ... but has a connotation of being a predator. "And the wolf went out on the dance floor."
This means clever and often (but not always) good looking ... but also not to be trusted. "And John came in, he was a real snake."
Inner city slang. Opposed to "Dog", "Dawg" means a friend who is loyal and sticks close to you, even if they're both referencing the same animal. For example, "How's it going, Dawg?" Note that you must emphasize the "a" or it could be offensive due to being misinterpreted as "Dog".
Someone who is uncouth, uncultured, and unwise, but also very obedient and loyal... if handled properly. E.g. "Fred's a real dog..."
Country slang. Means someone who is seen as high value and quality... to the right person, can be simultaneously complimenting and insulting. (Not to be confused with just "Pig" which is someone who is a slob.) For example, "Well, look at the prize pig up 'thar."
Someone who is messy and dirty and/or rude. Really not a positive connotation in any way. "He's a real pig." The rude meaning is almost exclusively used with men. HOWEVER... can also be a very rude way to refer to police officers. "Tony has the pigs on his back"
Someone who looks good and is a showman... but also not very complicated. "Wow, look at the showdog up there."
Believed by many to be a derivative of "Horse" (although possibly deriving
from a celebrity named Hoss), inner city slang. "Hoss" means a very attractive male who is likely well-endowed in bed. Like "Dawg", the "a" sound is emphasized. Pronounced to it rhymes with "Sauce" e.g. "He's a real haws." Downside: There isn't much else to them.
Someone who's fun... but can also be a minor problem. For example, "She was just a silly goose." It's a pretty regional term though. Also, not be confused with the act of goosing, which is pinching someone's butt. (Not to be confused with "Golden Goose" which other answers here mention).
This one is actually always bad. It means someone who takes from people constantly and never gives back.
People generally aren't called a bat, but they are called "Batty". It means kind of quirky, a little nuts, but generally harmless, and sometimes a source of amusement. "She's really batty." The one exception to not getting called a bat though is an older woman who can be called, specifically, "An old bat" which means about the same as batty or an older woman who is more emotionally cold.
Also, do not confuse with "Batman" which is a fictional character and being called "Batman" means you're being called rich, powerful, strong, larger than life, and there's a good chance it's not seriously meaning it. example: "I'm Batman"
Fat. Very fat. Also very insulting. Unless used in reference to online gaming, in which case it means someone rich who keeps free-to-play games free for everyone else by buying all the optional stuff. example for both: "She's a whale."
Someone clever, sneaky, predatory, and definitely out to get you, likely by scamming you in games of chance by having some unconventional skill like card counting. "Watch out for Lenny, he's a real card shark."
Someone who can't stand up for themselves, synonym with "spineless". Example, "Don't be such a jellyfish."
A Coward. Someone who runs at the first sign of trouble. Example, "What are you, Chicken?"
Fat. Not as bad as whale, however, unlike whale, doesn't mean they spend a lot at video games though. primarily used with women. Considered very rude, and would likely have nearby people get angry with you.
Alternatively, can mean an unpleasant woman.
Mindless and obedient, cannot think for themselves... however, used in a positive context in religion to... mean pretty much the exact same thing. Example, "They are a bunch of sheep."
Someone who is always underfoot trying to appeal to superiors. Generally considered to have no redeeming qualities other than being an easy pawn to use who knows they're a pawn (and potentially dangerous if they see an opportunity to grab power for themselves.) "Lenny's a stooge and a real worm."
A unicorn can reference an ideal person for a certain situation that doesn't exist. For example, a business wanting to hire someone with 10 years of a experience with a program that's been out for 3 years, they're looking for a unicorn. (And if they do find such a person, such as the person who wrote the program and spent 10 years doing so before release, that person is a unicorn.) Example, "That last candidate was a unicorn. Don't lose them!" or "Who you're looking for a unicorn, you'll never find them."
Possibly starting as a derivative of the above, in the lgbtq subcultures, a unicorn means a sexually attractive and active woman who is bisexual and polyamorous. Or even simply someone single, gay, and available. Generally denotes someone desirable who is non-monogomous. Example, "He's a cute unicorn." or "We're having a get-together of local unicorns."
Primarily means a sexually attractive and active man who enjoys many romantic entanglements. Can be either straight or gay. Generally denotes someone desirable who is non-monogomous.
Someone is is not sexually active or romantically fulfilling. Generally used more insulting or referencing an unfulfilling marriage. "Poor Susan, she married a real cold fish."
People aren't generally called cats, but they are called Catty. This means someone who thinks they're better than other people, and is quick to point out the failings of others. Example, "Hey, don't be catty."
Not to be confused with "Catgirl" or "Catboy" which is used as Anime subculture slang to refer to someone who is playful, flirty, and cutely attractive.