Casiopea said:He has lived there since when?
=> exact date/time questioned
Since two months ago.
=> exact date/time mentioned is given and expounded on using "ago" to make it more exact.
Consider:
Pat: He's lived there for two months. (non-exact date/time)
Max:Sorry? Since when, did you say? (exact date/time questioned)
Pat: Since two months ago. (exact date/time previously mentioned)
'two months ago' isn't what we would consider an exact date/time, but given the context, the pragmatics, it's as exact as it's going to get for that particular context. ;-) It's exactly what Pat said.
I still have to say that in any context the speaker and the listener will know what "two months ago" means. It's now May. Can we consider March an exact date or an exact time? No, we can't. I think a speaker is more likely to say the name of a month. If March means two months ago, then I think it's best to say "March". "Since two months ago" sounds a bit iffy to me. I think Cambridge University came up with a rather clumsy sentence transformation. They should remove it.
Why would Max necessarily say "since when" while Pat said "for" as in "for how long - for two months"? I don't think that's really predictable. I think Pat would more likely say "Sorry? For how long?" or "How long did you say he's lived there for?" "How long did you say he's lived there?"
I think Pat would most likely give the name of a month. I wouldn't rule out "since two months ago" as a possibility, but I wouldn't actively promote it - at all.