I booked a rewards ticket for my mother-in-law to RGN, it is composed of ANA, Thai, and Eva flights. We were looking at the booking and noticed that the DOB was entered into UA's system as May 9th (5/9) instead of September 5th (9/5). A call to UA was able to fix it, and said everything was all set. I checked the booking on Thai and it is still showing up as September. The flight is in January, and I'd rather not go through and cancel and rebook if I can't get this sorted.

Any suggestions on what my next step should be?

You'll be fine. They can correct it at check-in when they input the passport information.

And that will fix the rest of the flights?

It may or may not. But, whenever she checks in they will confirm the details that they care about for that segment of the journey. IIRC it's really only the US TSA/DHS that cares about DOB and even then I don't think they will reject anyone for booking a ticket with the wrong DOB (as long as you can produce the correct one on-demand). I would think that incorrect name spellings, DOB, nationalities, etc are fairly common I think especially since a good number of travelers don't do their own bookings. My firm's own travel agency usually just makes up DOBs for folks if they aren't provided and I've never heard of anyone having a problem.

My partner has had my birth year in his UA profile since 2012 and we've never had anyone even mention it at check-in, immigration, or security. I wouldn't worry about it.

The only practical impact of having the wrong DOB is that you might miss out on TSA PreCheck if you were otherwise eligible (which doesn't apply to non-US airlines anyway).