Learn English – What does a “meteoric rise” imply

Simile: He ran as fast as a tiger.

This is indeed a simile.

Metaphor: The tiger ran by me. (referring to a man)

This is not a metaphor, it is a different figure of speech called hypocatastasis.

Questionable: He ran with the speed of a tiger.

This is not a figure of speech, it is a literal statement.

There are really three common figures of speech of comparison (although the last is not well known by name it is really common in practice.) They are, in ascending order of strength:

• "simile" characterized by like or as
• "metaphor" characterized by the verb "to be"
• "hypocatasasis" characterized by the replacement of the word by that which it is compared to.

I remember the Mom telling her kid to clean up his room:

• "Your room is like a pigsty, clean it up!"
• "Your room is a pigsty, clean it up!"
• "Clean up your pigsty!!"

(With apologies to our porcine friends.)

The short answer is that the "arc of the moral universe" has nothing whatever to do with "the evolution of justice". "Justice", in the 19th century original of the quote as in King's paraphrase, is infinite, a constant: the "justice" referred to does not evolve.

God has made man with the instinctive love of justice in him, which gradually gets developed in the world. But in Himself justice is infinite. This justice of God must appear in the world, and in the history of men; ....

Look at the facts of the world. You see a continual and progressive triumph of the right. I do not pretend to understand the moral universe, the arc is a long one, my eye reaches but little ways. I cannot calculate the curve and complete the figure by the experience of sight; I can divine it by conscience. But from what I see I am sure it bends towards justice.

(From Ten Sermons of Religion, Theodore Parker, 1853, pp. 84-5.)

In context, it is apparent that "moral universe" is a metaphor for human history. The 'arc', then, is a rising curve, representing the "triumph of the right", the human ascension to heaven and human participation in the infinite justice of God:

That King was quoting an aphorism that arose from Parker's original use, if not paraphrasing Parker directly, seems well-established by the series of attestations of the phrase in various forms given by Quote Investigator: Exploring the Origins of Quotations, in the article titled "The Arc of the Moral Universe Is Long, But It Bends Toward Justice". As mentioned in the Quote Investigator article, a use of the phrase by King in 1958 shows King himself quoted the phrase:

Evil may so shape events that Caesar will occupy a palace and Christ a cross, but that same Christ arose and split history into A.D. and B.C., so that even the life of Caesar must be dated by his name. Yes, “the arc of the moral universe is long, but it bends toward justice.” There is something in the universe which justifies William Cullen Bryant in saying, “Truth crushed to earth will rise again.”

(Quote from The Gospel Messenger, "Out of the Long Night by Martin Luther King, Jr.", 8 Feb 1958.)

An image from that source:

The sense of 'arc' in King's use is metaphorical; the entire phrase might be restated as "[divine] justice will triumph in human affairs".