Related to the claim that there are only 4 non-kosher animals with one kosher sign and all of them are listed in the Tora is another claim: That all existing kosher animals are listed in the Tora.
In the Tora, we are given the signs of kosher animals (split hooves and chewing the cud). Most people agree that it is enough for an animal to have the two signs and that there is no need to link all animals back to the list, but some disagree (R. Mordecai Kornfeld, Insights into Chullin 59, section 6: The need for a mesora with regard to a chayah). In any case, despite the signs, the Tora does give a list of kosher land animals, but never defines it to be exhaustive. Here are the relevant verses:
These are the beasts which ye may eat:
the ox [shor],
the sheep [seh ksavim],
and the goat [seh izim],
the hart [ayal],
and the gazelle [tzvi],
and the roebuck [yachmur],
and the wild goat [akko],
and the pygarg [dishon],
and the antelope [te’o],
and the mountain-sheep [zamer].
Ox, sheep and goat are clear. Ayal probably is deer (Dr. Moshe Raanan: Ayalah Sheluchah). There is a discussion if tzvi refers to gazelle or ibex, the consensus seems to be that it is the gazelle (R. Mordecai Kornfeld, Insights into Chullin 59, section 7: The identity of the tzvi). The identity of the last five species (yachmur, akko, dishon, te’o and zamer) is uncertain (Certified Kosher).
Some people say zemer refers to giraffe which is kosher (R. Yirmiyahu Ullman: Kosher Giraffe). Another kosher animal is the American buffalo (or bison) and some people attempt to connect it to the te’o or yachmur (R. Ari Z. Zivotofsky: Kashrut of Exotic Animals: The Buffalo).
Caribou and elk are kosher, but I haven’t found anyone linking it back to any of the ten species on the list. Okapi, pronghorn, musk deer may also be kosher, but there is little information, also no linking to the list. There has been a debate about zebus in Israel. They are unquestionably kosher, but some people wanted to ban the meat because of a missing mesora (tradition) – which would mean it is not on the above list (R. Yehuda Spitz: Buffalo Burgers and the Zebu Controversy, R. Natan Slifkin: The Zebu Controversy).
As a conclusion, the claim that the Tora list contains all kosher animals is questionable and is not very convincing.