Seems like a British English vs American English distinction.
"It was pouring with rain" is understandable but does sound unusual to a native AmE speaker.
What is "it"? The sky, or perhaps the weather.
"It is pouring outside" clearly means "rain," and there's no need to explain "with what?" or "what is pouring?" because there is only 1 possible answer.
"It [the sky] is pouring [down] ... rain."
"Pouring" is a transitive verb - there must be a stated or implied direct object being poured. The preposition "with" does not modify either the object (rain), or the verb (pouring), so it's unnecessary and can be omitted.
If you are pouring water into a glass, or onto a potted plant, are you "pouring with water"? No, that's weird. You're merely "pouring [the] water."
"With" does suggest a logical, spatial relationship between the rain and the sky, but that's also obvious and could not be otherwise, so it's still unnecessary.
The sky pours rain. It does not ever "pour" anything else. It could "pour [rain] with gusto" or "pour [rain] with fury," but it still only ever "pours" rain.
Sometimes it is said to be "pouring buckets [of rain]."
Sometimes "it's raining cats and dogs," but it's never "raining with cats and dogs."
What other verbs can the sky perform? It could "dance with light" or "buzz with insects" or "be filled with clouds or stars," ... so in that sense using "with" seems somehow grammatically consistent ...
But in these cases the preposition "with" is necessary to explain the logical, spatial relationship between the sky, the object, and the verb. Insects and light do other things in other places. Clouds or stars are only found there, but the sky can "be filled with" many things so in those cases "with" is necessary to specify the object being discussed. If the sky is pouring anything at all, it's pouring rain.