Note: "Element" is used to refer to both individual elements and elemental pairs.

Element 0:
Default (Elementless?)

Element 1:
Zapper has attacks that use this element.
Unused armor from chapter 2, "Sky Mantle", has 50% reduction against this element.
"A cape that shimmers fluorescently. Protects against Elec and Holy attacks."
This description implies that this element is called Elec/Holy.
Spamton mentions "Thunder/Light" in the Sweepstakes, which seems synonymous with Elec/Holy.
Many recruits are ELEC element, including Zapper. Miss Mizzle's recruit description mentions a HOLY element.

Element 2:
Armor that provides resistance against element 9 is coded to additionally provide resistance against this element, as well as element 8.
Speculation: since the previous element is ELEC and the next one seems to be WATER or maybe ICE, this may be FIRE or WIND to form a set of classic JRPG elements.

Element 3:
Elnina has attacks that use this element. Specifically rainwater-themed bulllets.
Elnina is WATER:ICE element, according to her recruit information.

Element 4:
No information whatsoever.
Speculation: 4 means "death" in Japanese, so this could be Death/Scythe, as mentioned below.

Element 5:
Confirmed to be Dark/Star.
Used by the Knight, Tenna's star-shaped bullets, and some of the weather duo's attacks in the Party Dojo. (Not their attacks in chapter 3.) I believe it's specifically their sun-shaped bullets, but I'm not good at reading the code.
Autistic hair-splitting detail: Gerson's star-shaped bullets check if Susie has the Shadow Mantle equipped and reduce damage if so, but are not actually Dark/Star element. Likewise, the Titan and Titanspawn's attacks check if the Shadow Mantle is equipped and reduce damage if so, but are not actually assigned to this element. In both cases, it reduces damage by a different % than the Shadow Mantle normally does, so that's likely why the bullets are not directly assigned elements
Black Shard and Twisted Sword do more damage against "Dark-element" enemies. This is done in a similar way as the above check, where the game just manually checks if Kris has either of these specific weapons equipped and changes their damage against Titans and Titanspawn accordingly.

Element 6:
Confirmed to be Puppet/Cat.
It was speculated to be cat-related because the element is shared by Spamton NEO, Tasque, and Tasque Manager. Spamton confirmed the name in the Sweepstakes.
The cat Shadowguys have this element too.

Element 7:
Unused armor from chapter 1, "Mouse Token", has 50% reduction against this element.
"A golden coin with a once-powerful mousewizard engraved."
Speculation: Possibly MOUSE element, as it is next to the CAT element. Mice are symbolically linked to hands, see the Maus control and the basement angel doll scene. Hands control puppets.
https://toby.fangamer.com/interviews/nine/
Many anons have brought up this Undertale bonus content, where the number 7 is identified as "dog or wolf"-like, as possible evidence that element 7 is DOG. This is more recent than chapter 1 (and chapter 2), so maybe the MOUSE element was shuffled into another slot at some point in development. Some suggest that the pair could be DOG/MOUSE.
The weird route has a lot of Maice. Also... mICE. So the pair could be ICE/MOUSE. The Mouse Token references Chuck E. Cheese ("...from the family entertainment center?" - Noelle) which ICE-E's P"e"zza is also a reference to (by way of Five Nights at Freddy's.) ICE-E's Japanese name, Bulldokkun, indicates that he is supposed to be a bulldog, so this might also explain the speculated DOG connection listed above.

Element 8:
Armor that provides resistance against element 9 is coded to additionally provide resistance against this element, as well as element 2.

Element 9:
Armor that provides resistance against element 9 is coded to additionally provide resistance against elements 2 and 8.

Element 10:
Armor that provides resistance against element 10 is coded to additionally provide resistance against all other elements (except 0, the default element.)

UNSORTED:
Spamton mentions a Death/Scythe pair in the Sweepstakes.
Speculation: This is probably 9 or 10 considering protection against Death would probably give you protection against other things too. May be 4 if Toby is just a weeb.
Susie's Rude Buster is described as doing "Rude-elemental damage."
Kris and Susie's Red Buster is described as "Red damage."
Noelle's IceShock is described as "ICE damage."'
There are many elements only seen in recruit info, at least so far. However, Cat was first seen as a recruit element for Tasques and Tasque Manager before element 6 was confirmed to be Puppet/Cat. So some of the recruit-only elements may be part of the numbered elements.
Speculation: I personally think that ICE and BLADE, a recruit-only element, are a pair. In the sword route, Kris gets the title "Ice Blazer" with the description "Specializes in sword attacks." ICE and BLADE are also introduced together, as Rudinn Ranger (the first BLADE element enemy) and Head Hathy (the first ICE element enemy) appear in Card Castle sequentially. Additionally, it seems thematically appropriate for these elements to be linked considering the SWORD route's connections to the weird route.
Speculation: The order of elements 1-5 may correspond to reverse Undertale order. Elec/Holy corresponds to the Core and New Home. Element 2 would correspond to Hotlands. Element 3 is likely Water, so it corresponds to Waterfall. Element 4 would be Snowdin (Ice/Blade?) Dark/Star corresponds to the Ruins.
Speculation: Water/Music seems like a likely pair. Shyren, the song from the sea, Riverperson's singing, etc.

LIST OF RECRUIT ELEMENTS (SORTED ALPHABETICALLY):
BLADE
CAT
CHAOS (Jevil is listed as CHAOS:CHAOS in unused recruit info. Pippins and Shadowguys are CHAOS, too.)
COLOR
COPY
DUST
ELEC
FIGHT
FIRE
FROG
HEART
ICE
JEWEL
LUCK
MAGIC
MOUSE
MUSIC
ORDER
PUZZ
RABBIT
SMELL
SPIRIT
STEEL
WATER
WIND
VIRUS