The Chess960 analogue in shogi, xiangqi, etc Fischer said

In chess so much depends on opening theory, so the champions before the last century did not know as much as I do and other players do about opening theory. So if you just brought them back from the dead they wouldn’t do well. They’d get bad openings. You cannot compare the playing strength, you can only talk about natural ability. Memorization is enormously powerful. Some kid of fourteen today, or even younger, could get an opening advantage against Capablanca, and especially against the players of the previous century, like Morphy and Steinitz. Maybe they would still be able to outplay the young kid of today. Or maybe not, because nowadays when you get the opening advantage not only do you get the opening advantage, you know how to play, they have so many examples of what to do from this position. It is really deadly, and that is why I don’t like chess any more.

and

[Capablanca] wanted to change the rules already, back in the twenties, because he said chess was getting played out. He was right. Now chess is completely dead. It is all just memorization and prearrangement. It’s a terrible game now. Very uncreative.

and (emphasis mine)

I love chess, and I didn't invent Fischerandom chess to destroy chess. I invented Fischerandom chess to keep chess going. Because I consider the old chess is dying, or really it's dead. A lot of people have come up with other rules of chess-type games, with 10x8 boards, new pieces, and all kinds of things. I'm really not interested in that. I want to keep the old chess flavour. I want to keep the old chess game. But just making a change so the starting positions are mixed, so it's not degenerated down to memorization and prearrangement like it is today.

2 Questions: Does shogi, xiangqi or other co-hyponyms suffer any of the same problems that chess does, according to Fischer, which led him to invent chess960? Are there any analogues of chess960 in shogi, xiangqi or other co-hyponyms?
Notes to explain what I understand and to hopefully further clarify this question:
Note 1: I mean I really don't see any difference. They're abstract strategy mathematical board games that have pieces and stuff and there's always the same starting setup, soooo...
ETA Note 1.1: There seems to be an answer about how xiangqi is indeed like standard chess and yet another an answer about how shogi isn't. Most of what I know about shogi is from Shion no Ou, but I really do not see how standard chess is unique in this or at least how shogi isn't the same this regard. Again, they are all mathematical and abstract strategy games that involve pieces on a board and have the same starting position of the pieces. I believe it's the same with draughts/chequers. In each game, opening theory is I believe not only extremely viable to study but will dominate a very large portion of what a player ought to study.
ETA Note 2: Actually, I think the same problem will befall, for example the chess variants closer to the standard (in terms of the pieces used), such as 4 player chess, bughouse, crazyhouse, torpedo chess (the torpedo pawn thing), '5D Chess with Multiverse Time Travel': The initial setup is always the same, so it seems like, eventually, it's just gonna be about openings. Of course, each of these chess variants can be fixed in this regard by doing chess960 type shuffles or something.
For shogi and xiangqi, the answer is no. There is no analogue for western Chess960 in those chesses. Disclosure: I'm a decently strong player in all three.
The problems Fischer identified in those quotes you gave are that western chess favours the player with more opening preparation, and that there is no room for creativity. (Bear in mind that this is his own view as an insanely strong player who had reached the top level of the game back then.)
To answer the question from a broader context, I will give a little bit of history and cultural context to show why shogi and xiangqi, despite also suffering from the &quot;dogma of opening theory&quot;, never arrived at randomisation of the start position as a solution - indeed, the opening theory is not even perceived as a severe problem.
(Also, I will keep the context to professional/master players; at the amateur level, really anything goes for openings in all of the chesses.)
Shogi
Opening preparation is indeed important at the top levels. Historically the game has progressed from slower, cautious buildups favoured in the 17th-19th centuries, towards the more modern, faster game of the mid-20th century to today.
There are very sharp openings where one/both sides start attacking early and both players must walk tightropes (e.g. New Rapid Ishida; yokofudori/Side Pawn Picker; some lines of Gokigen Central Rook).
There are openings that have been examined to great depths (60 ply, even 80 or more ply) and involve many move order subtleties (Double Yagura games; Bishop Exchange; One-turn-loss Bishop Exchange).
There are openings which have gone even further, to survive in the ecosystem around it. (Fujii System was analysed literally all the way to mate in some lines by its inventory, Fujii Takeshii 9-dan.)
But yet, in the middle of all this, there is still room for creativity and variety. By the 1990s, of the two main styles, Static Rook was seen as the superior choice over Ranging Rook. The great player Habu Yoshiharu sparked something of a renaissance by playing Ranging Rook as an upstart young challenger in a title match - and winning. Since then different Ranging Rook styles, and even double Ranging Rook games, have never truly fallen out of favour at the professional level, even when engines overtook humans and universally preferred Static Rook.
There is more than enough to analyse, and still more creativity to be had (especially in double Ranging Rook games) - there was not a need for randomised shogi. This is of course not even mentioning arguments from tradition - there is no way that one can redo national culture by randomising the start position.
(Oh, and yes, randomised shogi would heavily disrupt existing opening theory, at least with regard to concrete moves.)
Xiangqi
Even before I talk history, the simple logistics of it is that xiangqi pieces don't randomise well. The kings, advisors and elephants are bound to the few points they can leap to, and are almost worthless if randomly shuffled away; the cannon movement means that they also don't randomise well, since on the first move already captures may be possible.
Now the history. Again, opening preparation is important at the top level. This should be no surprise again, as top players do like to seek any advantage they can get.
In ancient times (think 800 years ago), xiangqi was thought to be a win for red, the first player - classical manuals do some analysis on opening traps and games for red to win with central cannon versus screen horses, or opposing cannons (either one).
Fast forwarding to more modern times, there was the same sort of status quo from chess and shogi at one point: it was thought that black had trouble neutralising red's initiative, and the only defence against a central cannon was screen horses. The theory debates in this flowed from the 1930s onwards in national tournaments in China, with multiple red and black setups tested in battle.
In the 1960s, Grandmaster Hu Ronghua decided to revisit a &quot;busted&quot; opening. Was the Fan Gong Ma defence for black really losing? With tons of effort into research and with sheer playing strength, GM Hu Ronghua managed to bring the dead opening back to life, winning big tournament games with it and drawing other masters to try it. Again, this brought forth another period of experimentation that has not quite died down; even today, masters are exploring more than ever other red openings than a central cannon, with more fertile ground waiting to be explored.