What are some similarities between The Hobbit and The Lion, the Witch and the Wardrobe, or The Hobbit and The Philosopher's Stone?
Interesting question! I am going to deal with your first comparision that you suggest rather than focussing on comparing this novel to Harry Potter and I will focus my answer on one of the key differences in my mind between these two novels which concerns the presentation of christianity and their allegorical nature. Of course, there are lots of opinions out there but I don't necessarily think that Tolkien intended to use this novel to present Christianity in the same way that Lewis did with The Lion, The Witch and the Wardrobe. Of course, reading this latter novel clearly presents us with an allegory of Christianity - Aslan, as you identify, represents Jesus who willingly gives his life up for someone unworthy (Edmund) and then is resurrected as a result to defeat the forces of evil, encapsulated in the white witch. However, if we consider The Hobbit in these terms, there are no easy allegories that stand out. In fact, it is important to remember that Tolkien himself deliberately eschewed allegorical readings of his work. Thus although he was a Catholic and an important friend of C. S. Lewis, it is important not to try and read too much into his work and come up with meanings that are not there.
The one theme that is similar to both is the theme of good vs evil and how it is resolved. Both then present us with a simplistic world in which characters are easily slotted into one of two categories: good or evil. Thus dwarves, hobbits, men and elves are good and goblins and Wargs are evil. There is also the eventual triumph of good over evil. However, in The Hobbit there is a blurring of these boundaries, for example consider Thorin's questionable actions and decisions when he is suffering from gold greed or The Master of the Lakeland People. Also, you might want to think about the whole pretext of the adventure - robbery. Even though Smaug took the treasure from the dwarves originally, he had added to it from other sources, which the dwarves are planning to "appropriate" into their own treasure. Somewhat questionable.
So, in conclusion, I don't think we can treat these two novels in the same way. One is clearly a Christian allegory, whereas, to my mind at least, Tolkien seems intend on creating a mythical world loosely based on Norse and Anglo-Saxon mythology and peopling it, creating an updating of so many of these old myths and a compelling adventure story in his own fantasy world - quite an achievement!