Has anyone booked an apartment with this agency (All-Luxury-Apartments)?

I've never heard of this company, which means nothing. I did a Google image search for a couple of the apartments noticed that several of their apartments are listed also at Welcome2France and the about us sections of the two sites are basically the same with some tweaks to the language.

never heard of them either, and agree that that means nothing...

so, do the apartments listed all prominently display a Paris registration number? Different ones for each?

That's a start... assuming you are looking in Paris. You don't say

There are some rental exceptions for ground floor apartments and for rentals by the month, but for nightly rentals, online listings must display the 13 digit city registration number. For Paris apartments, these numbers begin with 75.

Sorry if I missed them, but I did not see any registration numbers. I would approach this company with caution.

Thank you for the kind responses. Indeed the lack of registration numbers is a concern — I thought it had been this site where I’d read something to the effect of “all our apartments are legal, we will provide registration numbers upon request” however I cannot find that on the site now so perhaps it was a different one where I saw that. I also don’t like that there are no reviews on the properties themselves. Oh well, it was worth an ask here.

I stumbled on to this website via a link from an article on blog of a gal living in Paris that I’ve followed for some time. The article was a few years old, so I emailed her inquiring what she knows of this company. She just replied that when she dealt with them for the article (not as a renter), that they were very professional and above board, however she’s had no contact with them since, so…. Times, they have changed so who knows.

In this situation I’d simply have moved on - it’s just frustrating because there is a property that is perfect for my needs this trip, and I’ve actually been in it. Somebody I knew stayed there in 2007 and we’d been in Paris at the same time so I’d visited her there. I recognized it instantly in the photos. Sadly, she is no longer with us so I cannot contact her and ask if she’d rented from this company.

Often landlords post their places on more than one platform, however, scammers also lift information about real apartments and advertise those and then steal the money. I'd never rent without cross checking and avoid places without reviews.

Yeah, agreed, Janet. I moved on. I hadn't heard of them and it seems nor has anyone else. I only gave it thought because of where I found it -- on the blog of a long-time Paris blogger that I have followed and know to be reputable. However once I contacted her and inquired, she admitted that it had been a few years since she'd had contact with them. C'est la vie.