Are airlines required to give the safety briefing in another language if asked?

Airlines are required to make announcements in English. When the departing has a different official language than English, they also make the announcement in that language at least. Sometimes they also make announcements in an official language of the destination. Some countries have several, up to 4 AFAIK, languages but I have never heard them make announcements in all 4 plus English.

It does not matter if the airline is based in the US. It must comply with local regulations, so if one end of the flight says that announcements must be made in a specific language, they do it. Only English is a requirement for all flights. Some people find it silly that given a flight between 2 non-English speaking countries, they still announce in English but having a common language greatly helps.

You can ask the staff to tell you what was announced in another language but they are not required to do so. Many flight attendants usually have pins telling people which language they speak, so that you may ask if you find one that speaks a language you understand.

From IATA (so more international rules), I just find:

2.38.1 Safety Announcements

It is important that passengers understand the safety announcements made on board. Airlines should, therefore, take into account passenger demographics when determining the languages used for announcements by cabin crew and, where necessary, employ the use of translators or video. Announcements should be clear, well-paced and able to engage passenger attention.

So nothing about English or other languages.

Additionally:

In addition, the briefing cards should be designed to be understood by passengers who are totally unfamiliar with aircraft and safety equipment, and who may have a limited understanding of any of the languages used.

And people with difficult to understand order ("language barrier" [IATA]) should be reseated from emergency rows.

Source: IATA Cabin Operations Safety, Best Practice Guide from https://www.skybrary.aero/bookshelf/books/3368.pdf

14 CFR 121.571 defines the FAA's safety briefing requirement for scheduled air carriers. Passengers must be "orally briefed," but it doesn't say that passengers must understand the briefing.

AC 121-24C expands on the regulation and provides more information on what the FAA expects. It states:

The pretakeoff oral briefing should be given so that each passenger can clearly hear it and easily see required demonstrations.

But it does not say that all passengers must be able to understand it. It does, however, contemplate that language difficulties could arise when it comes to exit rows:

The information regarding exit seating must be printed on the card in the languages in which briefings and oral commands are given by the crew

They recommend that exit row passengers receive individualized briefings, and most if not all US airlines require that you speak English (or another language used by the airline) to sit there, so that you're able to understand instructions in an emergency.

A post on the blog AirSafe News expands on this:

This advisory circular also requires that an airline provide passenger briefing information in the languages used by the airline. However, there are no requirements that every passenger should be given an oral safety briefing in a language that is understood by that passenger

As a practical matter, many international airlines will provide the briefing in multiple languages (sometimes via subtitles on tv screens) to try to reach as many passengers as possible. If that fails, the safety information card provides much of the information in pictorial form. See our previous question In-flight safety instructions for deaf passengers.