Does "with the descriptive noun of other noun" count as a simile?

Solution 1:

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.)

Solution 2:

It is a simile. Wikipedia's article on similes says:

A simile is a figure of speech that directly compares two different things, usually by employing the words "like", "as", or "than"... Similes are sometimes made without using the words "like" or "as". This often occurs when making comparisons of differing values.

The simile compares two separate things, the man's speed and the tiger's speed.

Keep in mind that all similes are metaphors; they are a subset. (Cf. Similes and Metaphors - are similes a subset of metaphors?)