The classic Burma Shave product was a brushless cream. For a while, though, the American Safety Razor company was producing a Burma Shave nostalgia kit, this one , which included a puck of soap. A couple of months ago I got one on eBay for substantially less than that Amazon seller is charging, $9.25 shipped. According to the tin, the mug and brush are made in China, but the soap was made in the USA. It has the Burma Shave imprint on it, and feels like a tallow soap, although that is not always easy to tell. There is no list of ingredients. Anyway, it works quite well, a good, usable soap. No scent at all, which for me is better than too much even of a pleasant scent. The performance seems similar to vintage Williams, maybe not quite as rich a lather. Which brings up the question, what is this soap? I'm assuming that somebody made it for ASR, as I've never heard that they had a soap division. Is it re-branded Williams, perhaps a formula from a few years back? I don't believe these sets are still made, although there is plenty of old stock out there. Don't know what year they stopped making them though, even assuming I'm right. Was the soap ever sold on its own, or only as part of these kits?
Sorry, can't help ID the soap, but I ordered 3 of the kits back in February of '09, one each for my two sons and myself. Still have 2 of the pucks left, my oldest son used up all of his, then moved on to Tabac. Can't really say I'm too fond of the soap, but I do break it out and use it on rare occasions.
I had one of those sets that I got in 1998 or 1999. I used the brush until it fell apart in 2009, then moved on to a VDH, and later an Omega. I still use the mug, and it currently houses a puck of Col. Conk Lime. I liked the soap and continued to buy it as long as it was available. Back then, I usually rotated between Burma Shave, Colgate, and Williams...which was all I could find locally and for that matter all that I knew existed. I seem to recall that Burma Shave was right between Colgate and Williams in the quality department. Colgate was my favorite soap from the time I started wetshaving (with an Atra Plus) in around 1992 until whenever it was discontinued. Burma Shave was a close second, and Williams was third (only because it was harder to lather than the other two) and I always got a great lather from all three.
Around four years ago my local Acme Market was still selling Burma Shave pucks...just the soap, not the kit. Then suddenly they went on clearance for 50 cents. I bought a few and liked the lather so I got some more from an online drugstore. I find the Burma to be better than either the current Williams or the final version of Colgate.