What's with Odysseus lying about himself?

My daughter (16) is reading the Odyssey. Normally she only reads fantasy, but reading Circe got her interested. I haven't read it yet, but will once she's done.

She was very surprised to discover that Odysseus arrives home on Ithaca with 200 pages left to go. She was also very baffled that he keeps meeting people who know him, then lying at length about who he is. In one scene he meets a shepherd who says he misses Odysseus and asks Odysseus where he is. Odysseus responds with 20 pages of lying stories about who he is, where he's been, and what he's done.

We discussed this a little. I maintain that Homer is enough of a writer to be doing this with a purpose, both the long stay on Ithaca before the end, and these liar stories. Eventually we decided that this seems to be humour. That the old Greeks thought it was hilarious to listen to Odysseus meeting people who love and miss him, and then misleading them with wild tales of stuff he's supposedly done. There is an earlier case near the start of the book that's quite similar, and that definitely did seem intended to be funny.

Thoughts?