Why is stigmata a plural of stigma?

When I first looked this word up on Dictionary.com, I found entries not for it, but instead stigma. I was baffled. Words in the English language usually follow the -(e)s pluralization pattern, but why not stigmata? Why can't this word be its own or an alternative singular? I know there are other plurals in English that follow neither pattern which people tend to mistake for singulars, such as taxa (for taxon) and strata (for stratum).

Words in the English language usually follow the -(e)s pluralization pattern, but why not stigmata? Why can't this word be its own or an alternative singular?

To be sure, the regular plural stigmas also exists, and Merriam-Webster indicates that that's the more common plural for the now-primary metaphorical sense "a mark of shame or discredit".

But as for why stigmata exists . . . stigma comes from Ancient Greek στίγμα (stígma), whose corresponding plural form is στίγματα (stígmata). Likewise for various other nouns in -ma, including schema (with plural forms schemata and schemas) and stoma (with plural forms stomata and stomas). (In Ancient Greek this was a very productive pattern; see Wiktionary's category for Ancient Greek words with the suffix -μα. Relatively few of those words have been borrowed into English, or at least their plurals haven't, so English doesn't have too many plurals in -mata; but the epenthetic -t- pattern is often reflected in related words: stigma and astigmatism, grammar and grammatical, asthma and asthmatic, symptom and symptomatic, coma and comatose, etc.)

Actually the plural form is stigmas, stigmata (the plural form in Greek) has survived with a different usage.

Stigmata: Etymology From Ancient Greek στίγμα (stígma, “brand”). Nominative plural: στίγματα (stígmata).

Stigma is a common word, but the correct rendering of the plural form of stigma can be confusing. We will examine the definition of the words stigma, stigmas and stigmata, where these words came from, the proper plural form and some examples of these words used in sentences,

A stigma is a shameful fact, characteristic or action that is attached to a certain person, circumstance or institution. A stigma causes disgrace. In medicine, stigma refers to a visible sign of a disease. The word stigma is derived from the Latin and Greek word stigma which means a mark made by a pointed instrument. Stigmas is the most common plural form of the word stigma, which is an Anglicized version of pluralization applied to a Latin word.

A stigmata is a repetition of the supposed crucifixtion wounds of Jesus Christ on saints and other Christian mystics, derived from the Latin and Greek word stigma. The term first appears in the 1600s. Stigmata is the correct Latin plural form for the word stigma, but the word is almost never used in this manner anymore. Someone who suffers from the appearance of the stigmata is called a stigmatist.

(Grammarist)

As said in the other answers, this is because the noun is Greek, not Latin. There are many such nouns in Greek, mostly ending in the suffix -ma (which frequently forms result nouns) although most such plurals don't survive into English

The following is diving into the historical linguistics of this far more than is probably necessary, but here we go:

The question then, is why does the Greek noun form its plural this way? The suffix -ma (and other words with the same plurals in -ata) form a special subclass of neuter third declension nouns. The third declension consists of consonant stems, meaning that, etymologically, the stem of these nouns ends in a consonant. In this instance, that consonant is -t. As the nominative (subject), accusative (object), and vocative (talking to) singular ending for neuter third declension nouns is -Ø (i.e. there is no ending, the bare stem is used) and t cannot occur at the end of a word in Greek, it is lost

This results in the paradigm we see where we have a word that in most singular forms ends in a vowel, but whenever it takes a suffix, a -t appears after that vowel

This doesn't quite explain it though. We only see nouns like this with singular -a vs plural -ata, not with any other vowel e.g. -e vs -eta, why is that? And the normal declension for nouns with stems ending in -t has -s in the singular rather than dropping it entirely so what's up there?

Now, third declension -nt stems solve the last of these problems, as the masculine vocative singular (which in the third declension, always matches the neuter nominative, vocative, and accusative singular) ends in -n without the t present in all the other forms

On its own, this doesn't seem to help us because there's no sign of an n in the -a/-ata words until we take a step further back in time, before Ancient Greek, to Proto-Indo-European (PIE). This is the reconstructed ancestor of most of the languages of Europe, as well as much of South and Southwest Asia, including Greek, Latin, English, Hindi, Farsi, Russian, and many others. In some contexts, in PIE, certain consonants could behave like a vowel and form their own syllable (e.g. *m̥ is an m sound, but without any accompanying vowel, like you're humming), and one of these consonants that could be syllabic like this was n. Through careful comparison between related words in various Indo-European languages, we can see that in Greek, *n̥ regularly becomes a (this is why Greek has the prefix a- where most other Indo-European languages have forms with the n like in-, un- etc)

So, these -at stems, turn out to be a close sister to the -nt systems, and come from PIE -n̥t stems. We can also see that the Greek deverbal suffix -ma comes from a PIE form like -mn̥t, which also gives us the Latin deverbal suffix -mentum (also neuter) which in turn gives us (via French) English -ment