How can I learn to recognize intervals larger than an octave? I recognize them in the limits of a single octave, but make mistakes when hearing larger ones. Perhaps it's because I don't know songs with them.
Another way is to learn to play chords with extensions (i.e. Cmaj9, Amin11). The extensions go past the octave. You should try to play the original chord, then play the extended chord. Listen for the tonal difference and you will be simultaneously learning the intervals. I'm suggesting this because I'm assuming that (based on your question) you can already recognize the intervals up to the octave.
You could try listening in your head to where that upper note (or lower) wants to resolve in relationship to the first note.
If say you have the interval of a perfect 11th from D4 to G5 you could think of D as your tonic (Do in moveable Do) and that G (Fa) as the fourth scale degree wanting to resolve to F# (Mi). Once you identify where the second note wants to resolve, it's just a matter of doing the theory.
If it were a major 10th where the second note (F#5) doesn't really need any resolution in D major you could just listen in your head to how many steps it needs to reach that tonic (D5).
The same one trains to recognize intervals in general. Find songs that have those intervals and listen to them again again. Songs like this might be harder to find, but I'm sure there are songs like this.
Then, try finding some programs or apps for ear training and train specifically with the intervals larger than the octave. Ear Master is a program that has this feature.
Well I myself am always interested in hearing about new techniques but I think the main way is to just play two strings or two keys together over and over and develop an intuitive understanding that way.
The thing with octaves is that the farther away tones are, the less likely they are to sound like they clash. Within an octave the relative interval is important, but beyond it is more of an open-ear'd territory.