GURPS is a generic system with some of the most extensive customization imaginable, which is a double-edged sword because while it means you can make anything, it won't always be easy. The default character is a human (~6 feet tall, two arms with hands, two legs with feet, two eyes, etc) and everything is built off of that. It being generic is also double-edged, because it means that the GM needs to determine which rules and options they're enabling. For instance, there are like 6 magic systems, and I personally would recommend the Sorcery system, as it enables free-form casting.

For example's sake, the traits of the phoenix with fire magic would look something like this:

• Assuming the phoenix is smaller than a human, it takes Size Modifier -2 (~3ft tall). This means that it can fit through smaller gaps easier, and in combat that human-sized things are at -2 to hit it, and it is at +2 to hit them. It will also presumably buy its Strength down to about 6 to represent being weaker than a full-sized human (though this is by no means necessary).
• To represent the wings and lack of arms, we take Flight (Winged), which removes the arms, then Foot Manipulators, Bad Grip, and Ham-Fisted 2, to enable the phoenix to grab things with its feet, but takes -2 when attempting to swing a sword or catch things with its claws, and a -6 penalty when it attempts to do something that would require high precision like picking a lock.
• For the sharp beak and claws, we take Teeth (Sharp Beak) and Claws (Sharp) respectively.
• It pays no points for speaking the common tongue, and then takes Speak with Animals (Own Family) to represent being able to talk to other birds.
• To represent the flaming aura, it takes Innate Attack (Burning, 2 points of damage, Aura). You can set it to Always On if you don't mind setting things you touch on fire, but by default it's toggleable. You can also obviously increase the damage.
• For the fire magic, it takes Sorcery (Fire Only), as well as Sorcery Talent (Fire Only). The former determines how strong a spell you can cast, while the latter gives a bonus to all rolls to cast spells. For spells, there are then two options, known spells and improvised.
• Known spells are relatively strong, but restricted. The points cost of a known spell cannot exceed the cost of the phoenix's Sorcery (eg if it has 48 points spent on Sorcery (Fire Only), it can buy a spell worth up to 48 points). These spells are built as custom, fixed abilities, and are bought for 1/5th the usual cost. Only one can be active at a time (though if eg a fireball starts a fire, the fire keeps going), and they take 2 seconds to cast.
• Improvised spells are weaker, but flexible. The points cost of an improvised spell cannot exceed the level of the phoenix's Sorcery (eg if it has 48 points on Sorcery (Fire Only), as above, it has Sorcery level 7, and so can only buy a spell worth up to 7 points). They are designed on the fly and while you're technically supposed to do the math and work out the points cost, the GM can just rule if something is too powerful. As an example, the ability to light a flammable object that the phoenix is touching on fire would cost 1 point, as would creating a flickering flame to function as a torch. This also takes 2 seconds.

More system stuff:

• You more or less have to design things on your own. This gets easier with experience, but eg the bestiaries that are around are pretty small.
• The tone is described as "heroic realism", and works better in more grounded games. It does not scale up to something like superheroes very well.
• The combat is incredibly detailed and has tons of options, if you want to use them, such as feints, multiple attacks, hit locations (which do different damage depending on the damage type being used to target them), cover, etc. On the other end of the spectrum, I have run games where combat is adjudicated through a single roll.
• By default it's fairly lethal, with HP basically staying the same from chargen till end-game, though this can also be adjusted.
• No classes, everyone just picks skills as appropriate for their character. Again, the GM will have to make a list, though you can also use Wildcard Skills which are more broad. Eg instead of buying different levels of Broadsword, Lance, Shield, Savoir-Faire (Nobility) and Heraldry, the PC would just take Knight! Wildcard Skill and get every appropriate "knight" thing at that skill level.
• Defined disadvantages give fun character traits in exchange for more points to spend. Maybe the Phoenix takes Bad Smell to represent that it smells like brimstone and that makes people that meet it less likely to react well, or Kleptomaniac so that it must make a check every time it sees a shiny object to not attempt to steal it.
• The only book you need is Basic Set. If you want more resources for dungeon-crawly things there's the Dungeon Fantasy series (in particular 1, 2, and 3, plus Monsters 1). You'd also probably want a book or two depending on the magic system. For the default system, that's Magic, for Sorcery it's Thaumatology: Sorcery, plus Powers. Absolutely pirate everything.

Pathfinder is pretty close to what you want

Especially now that they've officially added Awakened Animals and Tsukumogami to the mix

I have a character all set up that's a psychic umbrella, one that's a Minotaur detective and one that's a skeleton looking for some meat to wear

It's a damn robust system

It seems like a system that abstracts a lot of stuff might work. Alternatively one that is very generalized in its rolls with lots of opportunity to adjust them, like a burning wheel extensively homebrewed? Balance probably a nightmare, though. Not that that it always relevant. If there exists a crazy system specifically about different forms, I do not know it.

Tangentially, I've been trying to work out a dismemberment and limb replacement for my bug rpg, so if anyone has a drive-by suggestion, do drop it, thank you.