IR remotes use many different transmission protocols and IR frequencies, so I cannot say anything specific about your remote.
Repeated button presses can be implemented in multiple ways (depending on IR protocol):
1) send the same button press (repeat same codes)
2) send short command signaling repetition (receiver should remember last valid signal)
3) send alternating codefor multiple presses (for simple example, ENTER button code could be 00010, then 00011, then 00010, 00011, etc. I had LG tv which similar type remote)
4) any other way
I have tested multiple remotes - each may be different in radical ways, but most follow NEC IR pattern.
Code you posted may not take into account this behaviour. I would suggest for you too look at the code, modify it so it print IR pulse ON-OFF times. Then try pressing the button multiple times and see on-off time pattern - use this to adapt code to your remote.
Scope is not necessary, Arduino is well suited to decode IR frequencies.