Huxley's chapter which includes the Solidarity Service is a parody of church revivals in which the emotions of the participants are worked up to the point that, in such a charged atmosphere--believe that they have experienced the Lord.
Often there are tambourines that are struck in rhythm by people, charging their senses so that they will "feel the Lord." With this electrifying atmosphere, people now feel unifies with other folks and gives them a sense of belonging, shutting out the darkness of aloneness.
I think we do have forms of the solidarity service from BNW, but not in formal settings. The problem with the solidarity service is that it brings people together using a false bond. Aside from taking Soma and experiencing base physical pleasure, the members of the solidarity group don't actually do anything with one another. They do not converse, so they have no real bond. I used to see this all the time at my own social gatherings in college (many years ago, mind you). People get together to drink, flirt, and talk about things of no import, but never really get to know each other or connect to each other. They gain a sense of belonging, but form no lasting bonds.
Interestingly enough, the parallel to "Our Ford" isn't so off the mark today. So many people do worship their cars, and in our country alone, it is difficult to get anywhere without a vehicle--especially if you live outside of the big cities where there is little to no public transportation such as busses, taxis, subways, etc.
It might also be said that Hollywood stars (strange as they are) are worshipped by many, as are professional athletes.
I know it's not exactly the same, but it struck me that in today's society people allow so many other things (sports, money, work, pornagraphy, addictions, pride, etc.) to come between them and whatever Higher Power they might believe in--God, Allah, Buddha, etc.
I think the reverse of your question is more to the point. The Solidarity Service is the reverse of what we had/have, part of the new worship of "Our Ford" that has replaced the worship of "Our Lord." In particular, it is a parody of the Catholic Mass. The 12 members present at the service are reminiscent of the 12 Apostles or perhaps the 12 tribes of Israel. The strawberry colored drink they pass stands for the chalice of wine that is used in the Mass, and the passing of the cup reminiscent of the Last Supper. The "hymns" they sing, hideous though they may be, are parodies of the hymns sung at various parts of the Mass.
Since the Mass has outlived the Solidarity Service :), I guess you could say that we still have it today, even though we had it before. It's purpose is the same as theirs ... community worship and community building, although it functions quite differently.
We do not have solidarity services, but we do have other type of services, which are supposed to have been replaced by solidarity services in the Brave New World.
Present day services that come closes to the solidarity services of BNW are the religious services. A direct reference to this is contained in conversation between Mustapha Mond and Max Bernard. For example:
All crosses had their tops cut and became T's. There was also a thing Called God. ... We have the World State now. And Ford day celebrations, and Community Sings, and Solidarity Celebrations. ... There was a thing called heaven.
We can see clear parallel between the way the solidarity services are conducted in BNW and the religious services of today. For example, Making of sign of T by president, music, solidarity hymn, dedicated soma tablet and loving cup of strawberry ice-cream soma passed around and drank with statements like " I drink to my annihilation", and prayers like "Ford we are twelve make us one".