Does "Chinese laundry" have any meaning other than a laundry staffed by Chinese people?
- It was used literally in the piece you quoted.
- Those various regional uses of the term come from the literal meaning being so common at one point.
Wikipedia redirects "Chinese laundry" to Chinese Laundries in North America, which states that
In the United States and Canada in the late 19th and early 20th century, the occupation of laundry worker was heavily identified with Chinese Americans.
However, a google ngram search has "Chinese laundry" far more common than any other immigrant population (such as Jewish, Italian or Polish) plus "laundry".
Why "however"? As explained in your quote, in North America in the late 19th and early 20th century, the occupation of laundry worker was heavily identified with Chinese Americans (strictly, with Chinese Americans, Chinese Canadians, and Chinese immigrants to those countries, both legal and illegal, but your quote is still close enough).
As such, we would expect an ngram search to show that phrase as far more common than any other immigrant population. Since the Chinese were so strongly associated with that trade, but it's not a Chinese word (which could mean we would only bother to point out if a laundry wasn't Chinese), when we would expect "Chinese laundry" to come up much more often than any comparative phrase about another nationality.
Now, while the phrase is both literal and to be expected given the history of Chinese immigrants to North America, this does not mean it is not without connotation. This is the real however here, as is the fact that the "Chinese" part is irrelevant to the literal meaning of the phrase you quote first.
The 19th century had America's first political panics about immigration. Like today, the response to immigrants varied according the the ethnic and religious background of the immigrants in question; white Protestants were more welcome than white Catholics who were generally more welcome than white Jews (but only generally, I've heard of families having changed between "Cowan" [Irish, generally Catholic] and "Cohen" [Jewish] depending on just which seemed the slightly safer at the time), were more welcome than Asians, who were more welcome than Africans but the Asians were more numerous than the Africans, particularly the Chinese.
And just as today the stereotypical figure of "the immigrant" in the mind of an anti-immigration American would seem to be a Mexican, so too would the stereotypical figure of "the immigrant" in the mind of an anti-immigration American would then have been Chinese.
Now, the laundries where not great places to work, and the reason they were so much associated with the Chinese was that Chinese people often had difficulties getting other work.
This meant that they were part of the Chinese being associated with poverty, with hard menial work, and with laziness. (Yes, it makes no sense that a group be associated with both hard menial work and with laziness, but the financially less well off group in any culture very often are, often even in the minds of the same people).
And similarly, just while criminal elements within an immigrant community will increase the animosity toward the entirety of that community, often disproportionately to the amount of said crime, so the Tong Wars and the illegal business interests of the Tong influenced the image of the Chinese community as a whole.
This meant that the Chinese were being associated with drug-dealing, prostitution, gambling, theft, trafficking of slaves for prostitution (this latter having a salacious quality that made it all the potent a target of gossip and newsprint), and later with murder and riots. (Both riots within the Chinese community and some very nasty anti-Chinese riots in San Francisco, that still ended up hurting the Chinese community's reputation, rather than that of the white people who were doing the rioting).
And so, the laundries became part and parcel of a package of mostly derogatory stereotypes, beyond what they deserved.
As anti-immigration sentiment rose, so too did restrictions on immigration, and hence so too did people who did not meet those restrictions, viz. illegal immigrants.
And with that the laundries' reputation grew to include a reputation for hiring illegal workers.
Later when the economy suffered during the Great Depression, laundry work became more desirable to a larger number of people. For this reason, what was once a target of scorn became also a target of jealousy as a whole category of jobs seemed to only be going to a particular ethnic group. And so further, the existence of Chinese laundries became associated with negative attitudes toward Chinese people.
Now, with that all in mind, let's consider again the phrase "You could create a Chinese laundry with that much [washing powder]".
Well, if you had enough washing powder to create a Chinese laundry, then you've enough to create an English laundry or a Ghanan laundry or an Indian laundry: There is no correlation between amount of washing powder used and ethnicity!
But historically, laundries where often Chinese and so hence the phrase.
The objection "Mum, you can't say that!" is because while this isn't explicitly racist, it does have all the connotations above, and so has racist implications, especially when it's not strictly necessary for the analogy to work.