Are the words "Aural" and "Oral" usually pronounced the same? Does it vary by dialect? Are there strategies that people use to differentiate them when listening to spoken English?
Since this question asks about whether people can tell the difference I am going to answer for myself and my dialect rather than referring to a dictionary.
I speak Standard Southern English English and I pronounce them the same as far as I know. I cannot tell the difference when I hear them whatever accent the speaker has.
There is one exception and that is that I sometimes hear aural with the same vowel as now. Sometimes it is clear they are deliberately pronouncing it that way to distinguish it but sometimes I am not sure if they know how to pronounce it.
I never hear them pronounced slightly differently.
They are “supposed” to be pronounced the same in almost all accents of English (the exception would be accents without the “north-force merger”). It’s not unheard of for people to pronounce them differently, but none of the possible ways of differentiating them has become recognized as standard: they all involve pronunciations that many would consider “mispronunciations”. And I don’t know of any dictionaries that mention distinct pronunciations for these two words.
As mentioned in David Robinson’s answer, aural is sometimes pronounced with the sour diphthong. The phonetician John Wells made a blog post mentioning the "ˈaʊrəl" pronunciation of aural, and the comments below it have some more discussion: "trauma", John Wells's Phonetic Blog.
From an American English speaker with the cot-caught merger, “aural” might be heard with the unrounded back vowel of “caught” (vs. a rounded vowel in “oral”). For anecdotal evidence, see the posts here: http://www.verduria.org/viewtopic.php?f=4&t=73&start=200 as well the comments left below your question by Hot Licks and choster.