How can I compare members from different dimensions that have the same key values?/Lets say I have dimensions [Delivery Date] and [Ship Date]. How can I select just records that were Delivered and Shipped the same day?

You can use FILTER function and compare member keys using Properties function:
SELECT {[Measures].[Internet Order Count]} ON 0
, FILTER( NonEmptyCrossJoin( [Ship Date].[Date].Children, [Delivery Date].[Date].Children
)
, [Ship Date].[Date].CurrentMember.Properties('Key')
= [Delivery Date].[Date].Properties('Key')
) ON 1
FROM [Adventure Works]