The remaster of MR1&2DX (Steam/Switch/iOS) does have a random button in addition to a searchable internal database of Albums, Song titles, and games.
For whatever reason, the Random button in Ultra Kaiju is absent from the North American version, so you just either have to enter in some random letters/words in the Keywords field, or use NFC. The Japanese and Southeast Asia versions of Ultra Kaiju have a CDDB and Random button, though the "CDDB" is a bit misleading because it actually uses the keywords lookup to generate the Kaiju from the CDDB entries. In this regard, at least, the North American version is superior since the Keywords aren't locked to a pre-defined list.
Why did JP and SEA get a Song List and NA got Keywords for UKMR? Maybe copyright or licensing, though it's honestly anyone's guess.
But it still would have been nice for them to keep the Random button :)