From what I've heard, Burma Shave soap is/was (they may still be making it, or at least they were until recently) pretty bad, and I haven't heard anything about it having been reformulated since it was introduced. The brush too supposedly is quite bad, it's got clipped bristles. That mug is pretty though!
There have been at least 2 formulations. I have 3 cakes in my vintage stock that list sodium tallowate as the first ingredient. Recently on The Bay I saw some for sale that is sodium palmate based. My boxes do not list metric weights, so they probably date to the early 90's or 80's.
I have not shaved with Burma Shave soap since the 90's, and I recall liking it. A Burma Shave brush, soap, and cup set was my first wet-shaving buy after breifly using shaving gels in college. I have no idea if the stuff I used was the talow or palm oil based. The brush was not good, but it got the job done. I used to just lather it on the puck with a wet brush, the way my father taught me. That thing would probably have made me bleed if I had tried to face lather.