The composition of pumice wouldn't be similar. But for horticulture, the actual mineralogy may not matter too much. Apparently, they are first made up of a type of rock, which says what combination of minerals are actually inside.
Akadame is not a soil or a pumice. It is volcanic ash and dust that has sedimented into a clay rock. And once sedimented, was eroded further by rain. And that after forming a clay rock, the minerals actually change to allophane and imogolite. As these are formed by either weathering or metasomatism. Making it a metamorphic rock.
I have seen kiryuzana being referred to as sand. Apparently, it is sand from the Kiryu region, 桐生砂, the zuna part meaning 'sand'. And it coming from Gunma Prefecture.
Looking at some Japanese site descriptions, this one seems to be pretty much the consensus:
"Kiryu sand is volcanic gravel that is produced from a layer even lower than Kanuma soil, and is mountain sand produced around Kiryu City in the Akagi Mountains in Gunma Prefecture. Kiryu sand is harder and more angular than Kanuma soil, and has excellent breathability and drainage, so it is mainly used for plants that do not like excessive moisture. It is often used for bonsai such as pine and cypress bonsai, so it is a must-have soil for those who want to grow pine bonsai. It is also often used for wild plants, and may be used alone for perennial plants.It contains a lot of iron, so it is reddish yellow-brown and slightly acidic. It contains a lot of iron, but it is clean and does not contain fertilizer. The grains are strong and do not break down even with repeated watering, so it can withstand long-term use. It is truly ideal as a basic soil for bonsai. However, Kiryu sand alone drains too well, so it is often mixed with Akadama soil. In addition to bonsai, this soil is also widely used for oriental orchids, omoto, wild plants, succulents, etc."
Not sure if they mean it is actually below the kanuma layer. Is there kanuma in the ground all across Japan? So they have kanuma in Kiryu city and kiryu in Kanuma city? But they dig one up ine one spot, not the other?
And would it be very similar to most types of lava one can buy? Often, lava is red as well, which is I assume also because of iron ixide. And it is also harder and much more dense than akadama.