This is another age-old issue. Training material should start with "cleared approach" as the most rudimentary form of an approach clearance, then work up through increasing levels of specificity from there. We did that in a live training course a few years ago and it was met with a room full of jaws agape.
Instead, the first exposure to approach clearances for pilots tends to be a PTAC for a precision approach, "[distance] from [reference], fly heading [intercept heading], maintain [altitude] until established on the localizer, cleared ILS RWY [rwy number] approach."
For this specific case, as crazy as it might seem, if the controller didn't specify the RNAV Y or Z, then 'cleared RNAV RWY 9 approach' means the pilot can fly either, despite what might have been _specifically_ requested earlier. What matters is the approach clearance, not the request. This is further mitigated by the fact that the controller's initial verification was, essentially, "you want the RNAV RWY 9, right?" which also didn't specify the Y or Z.
It's very common for tower to issue alternate missed instructions in the event of a missed in any case, so the published missed might not have been relevant, OR, the missed approaches might not vary enough to warrant a distinction as the radar controller could easily protect for both. In this specific case, the missed approaches vary radically between the two approaches, so, I can only imagine it was a slow news day and the controller was happy to protect for both, OR it was an oversight on the controller's behalf.
Lastly, it's frustrating that the CFII wouldn't have covered this as a practical matter during training. A DPE isn't likely to cover it, but if a CFII wants their candidate to fly with confidence in the system, matters like this should be covered during training. I get that not every CFII is going to know the 7110.65 inside and out, but having multiple RNAV approaches to the same runway is somewhat commonplace - this should be addressed during training, along with how to properly request the approach by name, "RNAV Yankee runway niner."
I disagree that clarification is required when receiving the clearance (unless it's because the pilot isn't aware of the guidance regarding approach clearances). Presumably, the controller knows there's multiple RNAV RWY 9 approaches and would've specified one vs the other if required. The fact that he/she didn't definitively means you can fly either.
This is in the same category as being assigned a speed restriction in the terminal environment, and then knowing that a subsequent approach clearance cancels any previous speed restrictions, unless a new speed restriction is assigned as part of the approach clearance. There's no need to ask. I'm all for seeking clarification where ambiguity exists, however, the approach clearance (much like the speed restriction example) is not ambiguous.
