Chemistry is somewhat environment- and temperature-dependent, but there's no other element that behaves like carbon in any known conditions.

Carbon's chemistry comes from three major factors:

(1) It forms four bonds readily. (2) It can form double- and triple-bonds readily. (3) It bonds strongly to itself in a configuration that allows it to form long chains.

(1) is satisfied by other elements in its column on the periodic table, but silicon (the next element down in its group) and the following members (germanium, tin, and lead) fail (2) and increasingly (3). Silicon will form chains, but is reluctant to form double bonds; in general, double bonds become weaker for atoms further down the table. Silicon (and silicone, chains of Si-O bonds) are the most promising analogs but they have a lot of problems.

(2) is satisfied by most other light nonmetals, but those nonmetals mostly fail (3) and almost all fail (1).

The highly-electronegative oxygen doesn't really want to bond with itself (failing 3) and almost always takes a -2 oxidation state (failing 1).

Nitrogen actually does form four-bond atoms decently often (most notably the ammonium ion), so it somewhat satisfies (1) to some extent, but is so eager to form N2 that most polynitrogen compounds are wildly unstable to the point that "nitro" is a term even laypeople know is associated with explosives (failing 3).

Boron can form three bonds, allowing some of the complexity of (1), and will form nice polyboron compounds (3), but doesn't like forming double bonds (failing 2), and the bonds in boranes are so weak that they're mostly quite reactive (weakening 3).

Sulfur will form polysulfur chains (the most common form is an eight-membered ring), but sulfur (like its cousin oxygen) usually doesn't form more than two bonds except with extremely electronegative partners (like its effective total bond order of 3 in sulfur dioxide or 4 in sulfur trioxide).

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There's another problem here too: life is likely to form out of common elements in its environment, and carbon is just WAY more common than the alternatives. The mechanisms by which the elements are formed in stars very strongly favors elements with even atomic numbers (because they are mostly formed from helium-4 nuclei) and the burning processes peak at carbon/oxygen, neon, magnesium, and silicon.

As a result, carbon is very common. It's the fourth most common element in the Universe (after hydrogen, helium, and oxygen), and ~an order of magnitude more common than any of the elements discussed above except oxygen (which has basically no analogs to carbon chemistry).