The documentation for the missingkey option says:

missingkey: Control the behavior during execution if a map is indexed with a key that is not present in the map.

It sounds like it might apply to index builtin function as well, but it applies only to the .X.Y.Z syntax.

https://go.dev/play/p/ZneXhxfIBWT

package main

import (
	"fmt"
	"io"
	"text/template"
)

func f(text string) {
	fmt.Println(text)
	t := template.Must(template.New("").Option("missingkey=error").Parse(text))
	err := t.Execute(io.Discard, map[string]any{})
	fmt.Println(err)
	fmt.Println()
}

func main() {
	f(`{{ .Foobar }}`)
	// template: :1:3: executing "" at <.Foobar>: map has no entry for key "Foobar"
	f(`{{ index . "Foobar" }}`)
	// <nil>
}

It's hard to say whether this is behavioural bug, or just an ambiguity in documentation. Although I have use cases where missingkey applying to index builtin would be preferrable (replacement for .X.Y.Z if map keys don't happen to be valid identifiers), this indeed would be inconsistent with index expression of Go language itself.