Most Americans (and nearly all Canadians) pronounce those two names exactly the same. Note that that’s nearly 400 million native English speakers. Not much compared to the population of China, but several times greater than the combined populations of England, Wales, Scotland, both Irelands, Australia, and New Zealand. Also note that most British speakers do not speak with “standard British” pronunciation anyhow. (BTW, I typed them in Google Translate, and it pronounced both exactly the same.) Additionally, note that many Americans also merge two or more of “Mary”, “merry”, and “marry”. The exception is the East Coast, whose vowel phonemes are similar to those of RP (so-called “standard British”). There are of course many mergers in the speech of British people, depending on the region. RP speakers often merge "gulf" and "golf", for example.

Yes, they are identical. The actual sound vowel sound will vary depending on the regional accent of the speaker, but most native speakers will say these names in the same way. 'Morrie' became popular, especially for Jewish men in the US and Britain, in the twentieth century. It is very rare today. If I heard that name these days, I'd presume it was someone reminiscing about their great-granddad in Brooklyn in the 1930s. I have never come across anyone called 'Maurie', but if I did, I would pronounce their name /ˈmɒri/, the same as 'Morrie', and rhyming with 'sorry'. In American English, most people would pronounce both names /ˈmɒri/ . A BrE speaker would have a pronunciation that's 'tighter' and closer to an 'o', while an American's pronunciation would be more open and closer to an 'a', but neither would distinguish between the two names. Why would I rhyme 'Maurie' with 'sorry' rather than 'glory'? Why a short 'o' rather than a long 'or'? After all, as Dan has pointed out, these are two different sounds in British English. Here's the answer - I'd do so because names have their own conventions. I'd follow the convention of similar names with 'aur' in them. For example, 'Laurie' in BrE does not rhyme with 'glory' - it rhymes with 'sorry'. Likewise, Laurence and Maurice (longer versions of 'Maurie' and 'Laurie') also have a short 'o' vowel, rather than a long 'or' vowel. Proper names, of people, places, companies and so on, often have their own peculiar pronunciations. Many names, particularly surnames and British town names, are often pronounced in a way which is impossible to guess from the way that they're written. (Note to Phil: There may be 400 million North American English speakers, but how many of them know how to say Leicester, Gloucester or Worcester?). You basically just have to 'know' how to say certain names, or, failing that, you need to ask someone.

Some people will pronounce them both the same, with a short "OR" sound after the initial consonant. But most people will pronounce them differently, as follows: Maurie has a long "OR" sound, like in "Boring". It sounds like "More-E". Rhymes with Glory, Whorey, Borey, Gorey, etc Morrie has a short "OR" sound, like in Lorry. It sounds like "Mor-E". Rhymes with Lorry, Sorry, etc Try typing them into Google Translate to hear the audio, which is accurate for these names.

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Pronunciation of each will vary. Sometimes they will sound the same and sometimes a little different. It will depend on how the person wants their name pronounced - some will want a slightly different pronunciation to others. As well, there can be a slight variation from one English speaking country to another as patterns of speaking are different.