Since many of the luxury properties in the Los Angeles area are also part of historic preservation and conservancy groups, it's important to know who's bought them. In the event of damage, the owners have to be willing and able to have the property restored according to very specific guidelines. The more people who know the owners, the more eyes can help fix problems before they become major issues. Also, because there's a lot a secured loan money involved in such purchases, Feds have the right to let other property owners know who's borrowed the publicly-financed money.
There should be no reason to hide from the government who is living where. Though property buyers should be recorded in a federal database, that doesn't mean that the information should be open to the public. There should be a balance between public safety and personal privacy by having an official record of property buyers that is only available to government agencies like police or child services.
When buying non-luxury properties, in most cases, people having to prove who they are. They are required to provide a social security number, undergo a criminal background check, and much more. The case should not be different just because there is more money involved. If anything, it should be more strict.
It would be too easy for people to spend illegally obtain funds to purchase luxury Los Angeles properties if they are not required to reveal their identities. Proof of one's identity is required in most aspects of daily and is the norm for financial transaction. Allowing anonymity in the purchase of high-end properties is an open invitation to criminals.
Given the reason they are doing this, it goes against the 4th amendment. For any given individual property owner they have no reason to suspect money laundering, asking for their identity is an unreasonable search and seizure.
However, if there was a federal property tax then it would be OK for them to get the information that way. That would not set any precedents weakening the 4th amendment, since then they would be requesting the information for the purpose of enforcing tax law, although they could also look over the information to discover evidence of money laundering.
Even then some of the properties may be in the name of a company but if something suspicious enough to warrant an investigation were discovered then they could ask for the identities of the people who carried out the transactions without raising any 4th amendment issues.