Why does 目 have a counting sense to it?

Question

Why does 目 get used in situations like 二個目? How did it come about getting the meaning of eye as well as some counting meaning?

Answer 1

I think it's related to the sense mentioned by Brandon, from an underlying idea of "something that sticks out visually", hence the use of 目{め} "eye". This same sense of "visually notable or prominent feature" gives rise to many other uses of 目{め}, such as 結{むす}び目{め} "knot" from 結{むす}び "tying" + 目{め} "visually prominent feature", or 籠{かご}目{め} "lattice" from 籠{かご} "basket, basket-style weave" + 目{め} "visually prominent feature". It also attaches to adjectives (though it is generally spelled in kana then), as in 長{なが}め "longish, on the long side".

FWIW, Shogakukan's entry for 目{め} breaks the definitions down a bit differently than Daijisen. Some of the definitions that are relevant to ideas of counting include (my additions in [square brackets]):

(3)　見る対象をいう。
Indicates the object of seeing/looking.

or:

(5)　連続する、物と物との隙間（すきま）。間の区切り。区切りをつける線条。また、そのように刻まれたもの。
Gap between successive items. Separation between. Line separating things. Or, a notch cut as such a line.

or:

(6)　空間的、時間的な切れ目。二つの物、あるいは二つの事態の区切りや接点。転じて、物の条理、また計量の区切りや、単位をいう。多くの場合、動詞の連用形と複合して用いられる。「切れ目」「切り目」「分け目」「折り目」「裂け目」「境目」「合わせ目」「繋ぎ目」「綴じ目」「くけ目」など。
A spatial or temporal gap / break. [Note that this definition uses the word being defined...] The divider or point of contact between two things or situations. By extension, a division or unit in the reason for or measurement of things. In most cases, used in combination with the ren'yōkei [continuative form, i.e. -masu stem] of a verb. Kireme "break/gap", kirime "cut", wakeme "split", orime "bend, fold", sakeme "rip, tear", sakaime "boundary, border line", awaseme "join, seam", tsunagime "connection, joint", tojime "binding", kukeme "stitch", etc.

or:

Ⅲ　〔接尾〕 (Suffix)
1　数詞のあとに付いて、初めから数え進んでひと区切りをつけた、その区切りまでの数を表すのに用いる。「二番目」
Attached after a counter [such as 個{こ} or 番{ばん}], used to express the number counted when counting the divisions between items from the start [that is, the ordinal or positional number]. Nibanme "second"
.
2　形容詞の語幹、動詞の連用形などに付いて、そのような度合、加減、性質、傾向の意味を添える。「細め」「長め」「控えめ」「おさえめ」など。現代では、形容詞の連体形に付けていう場合もみられる。「細いめ」「長いめ」など。
Attached to forms such as an adjective's stem or the ren'yōkei of a verb, adds a sense of that kind of degree, extent, quality, or tendency. Hosome "slender-ish", nagame "long-ish", hikaeme "moderate, with reserve", osaeme "restrained", etc. In modern Japanese, also seen attached to the rentaikei [attributive form, same as the dictionary form] of adjectives. Hosoi-me "slender-ish", nagai-me "long-ish", etc.

That's a lot to digest, so:

TL;DR:

The word 目{め} is generally translated just as "eye", but idiomatically, it is used to express a wide array of ideas related to the basic sense, "visually prominent feature". This is one of the more productive and useful words in Japanese, and learning the many uses of this term will serve you well.

Answer 2

Wild speculation, but it could be related to sense 10 in 大辞泉:

10 線状に1列に並んだものの間にできたすきまや凹凸。「櫛【くし】の―」
A gap or unevenness occurring between items arranged in a line. kushi no - [me] (the gaps between the teeth of a comb)

And from this notion, the notion of first, second, third. Or, maybe it's just 当て字 for some homophonous grammatical suffix. I couldn't track down anything concrete though.