Sorry to dig out this rather old thread, but I thought worthwhile just sharing on here my recent experience with the same issue (and apologies this got longer than I thought):
Booked CDG-SGN-BKK one way with Vietnam Airlines for September, in Premium Economy for me and +1. One detail is that the CDG-SGN flight is a codeshare with Air France (VN2106 op AF258). Pricing was EUR 805.00 one way, same as all the other options on VN metal (via SGN or HAN) the same day. Opted for the AF codeshare because (1) the timing was better and (2) Air France PE seems to be a superior product, so why wouldn't I take it for the same price? And direct through AF, same itinerary was something like twice the price (was priced the same as the nonstop CDG-BKK AF flight).
After booking, I got my VN booking reference and of course couldn't select seats or really manage anything on the VN site for the AF flight (unsurprisingly). Tried the AF manage booking function, but the VN PNR doesn't work with them. And no AF PNR provided by VN anywhere (@OP - how did you get your AF PNR? Or did the VN one just work? Do you remember?) So I started googling and came across this thread and now noticed fare bucket "U" on my ticket also, and alarm bells go off... (Was it too cheap to be true after all?)
Emailed VN online support last night around 11pm straightaway to ask for AF PNR so I could assign seats on AF, and to confirm cabin is PE all the way. To my surprise, I got a response within 20 mins (!!), no mention of the AF PNR at all but offering to assign seats on my behalf free of charge (even included a small snapshot of the seat map which was definitely PE with 2-4-2 seating on the AF A350. So they assigned us seats 12A and 12C. (Btw: assigning seats on the cheapest AF PE fare for this route would have cost me EUR 147.00 each, so somehow have I just saved $480??)
Then I noticed: hang on, row 12 is a different cabin layout than what I saw when I checked on AF beforehand (AF258 on the Air France page on my day of flying showed the newer layout with PE in rows 15-18, not the older one with rows 10-12 that somehow VN seem to work off for the same flight...how does that even work??). At this stage, not sure if I should be relieved or even more concerned - I have a PE seat allocation but in a seat that doesn't exist on that flight with the operating carrier, how is this going to end come check-in time?
So this morning, called VN and basically they can confirm as much as I can see online in their systems, I am in Premium Economy, fare class U is PE with VN, I am allocated seats as to what they can see the seat map to be for VN2106, and no there is no Air France PNR ("You booked with VN, what the heck are you talking about?" kind of attitude), and I would need to call Air France then for questions re Air France. Pointing out that I will need my AF PNR for that, they seem to realize "ahh yeah...hang on", put me on hold for 2 mins and come back basically just telling me to call them regardless, tell them your flight details and fare group U, and ask them to confirm which cabin that is. Thanks for that, I would have done that anyway...
So called AF right away, got a super helpful agent named "Jen", and once I mention VN codeshare she opens with "we get dozens of calls about this every day, you really need to work with VN to get the AF PNR from them..." - good start. So I tell her, all I really want to know is, according to AF am I in Y or in PE, mentioning the fare bucket VN booked me into is "U". Straightaway she says "well that's Economy, I am sorry". Thought so.
Then I mention the seat allocation for row 12, and she gets interested. Asks for my ticket number (which I have), and puts me on hold to "investigate with the backoffice". Comes back a few minutes later and confirms "yes indeed you are seated in PE", there are two different layouts etc. Why does the seat map not match - no idea but row 12 is definitely PE on the older A350 layout. And also I asked can she tell me the AF PNR - tells me there is none on that booking, only the VN one and that doesn't work with AF. Arrgh...
And that's where I am at... cheap VN Premium Economy fares operated by AF are not for the faint-hearted. I am still fearing, given the missmatch in seat maps and row 12 not actually being PE on the AF flight, come check-in on the day I'll have an issue as for AF this is still booked in U. Also, without an AF PNR I can't even do online check-in 48 hours in advance so have to wait until we are at the airport? A bit late for my liking to deal with this if it indeed goes pearshaped and we end up in middle seats down the back. If I would travel by myself I would just wing it from here, but with +1 tagging along I really don't want to risk it.
Next steps: probably trying to get some form of acknowledgement in writing from AF that my booking is indeed in PE, maybe via their WhatsApp? Any other ideas?
Thanks for reading and any encouragement/tips anyone might have.
Greetings from aboard AF258 / VN2106 ! Since my original post (quoted) a few months ago, the day of travel has finally arrived and I thought I'd post an update on how this all went down, given the concerns I had. For anyone in the future who might be in the same position.
[For a quick TLDR: booked a cheap VN codeshare op by AF CDG-SGN-BKK in W (the SGN-BKK leg is in Y with VN), for about 60% of what the AF ticket would have cost. We already had worked out that there were likely no issues with the ticket itself although originally there was concern about the fare bucket, but my seat assignment that I got free of charge through VN (while AF was charging 90 EUR per person) was in question as the two cabin configs didn't match. Consensus was "wait until T-48 and see what happens". So that's where we will pick up below.]
On Sunday afternoon T-48 rolled around and voila!, the AF app all of a sudden shows my booking (wasn't able to see it or add it before), most likely due to my Flying Blue number I had in my VN booking. And it shows a completely different PNR than the one issued by VN.
Now, checking seat assignments straightaway (as I wanted to make sure we had a window-aisle combo), it shows 12A (which in the AF app and config is a seat in J) and 12C (which doesn't exist at all so doesn't show on the map). Tried to do "change your seat" in the app but AF of course still charges 90 EUR each, so thought I'd try via chat. Agent can't help, due to VN ticket. Tried via WhatsApp chat, the AI bot cannot help either and then requesting an agent takes half a day. Also tried on the laptop via MMB on AF homepage, but cannot do anything on there, claims the PNR doesn't exist. So I decide to just wait for check-in to open at T-30.
In the meantime, I get an email from AF to enter passport details and upload Thailand Digital Arrival Card, which all works perfectly fine. When doing that I check the app again around T-28 and all of a sudden our seat had changed to 17G and 17H, in the centre block. Bugger!
So T-30 arrives yesterday morning and I open the app a minute after: "Please check in at the airport."

Try online in MMB: still doesn't even recognise the PNR. Then half an hour later I check my emails and see the "Check in online now" email from AF and think "yeah, I wish!", but click on the link button anyway and... it works!!??!! Yes, indeed via the email link I was able to check in and change my seats (now for free!), was able to snag 18K and 18L. Only thing was that they wouldn't issue boarding passes, but given we had to check luggage that wasn't a big deal. I had my seats and was checked in.
Now, there is one major (in my view, at least) downside with the VN codeshare to be aware of:
apparently it does not qualify for the usually very reasonably priced paid upgrades that AF offers. Nothing was showing up on the app, so I enquired via the online chat. Was told yes upgrades to J are indeed available for my flight, but if it doesn't show in the app they cannot help either and I need to wait until airport check in. Same response when I tried again later via WhatsApp.
At the airport, the lady at check in was very eager to help, confirmed yes, even now at T-2.5 there was still availability, but couldn't get the system to spill out a price. She tried to call ticketing and they couldn't help either, so she sent us to the ticketing desk around the corner. Two ladies there where at first baffled, but then concluded "because you're on a VN codeshare", they cannot sell me an upgrade. Reason being (I kind of understood between the lines), upgrade pricing is based on fare bucket and price paid for the original ticket, and the VN codeshare is a block codeshare and AF has no visibility of what I paid etc. (Not sure if this is true or not, but that's kind of the explanation I got.) In any event, they would all have happily taken my money but in the end "Computer says no!"
Bit of a bummer, but AF Premium is actually excellent so not getting the paid upgrade option helps with the finances, especially after 4 days of shopping in Paris

.
Okay, so in the end all is well but it needed active managing of the booking (which I do anyway).