IIRC, the same chrism used for people is used for altars and churches.
And...I'm not sure I would say the Holy Spirit is "in" the chrism in the same way the Son is in the Eucharist. The chrism is holy and carries with its anointing the "seal of the gift of the Holy Spirit" but such authority is given in truth only to the Apostles and their successor bishops. Originally, we see in the Book of Acts that for a community to be fully integrated into the Church by the reception of the Holy Spirit, at least one of the Apostles had to go and lay hands on them directly.
Of course, this devolved to bishops later on and chrism was used very early as well (I believe Tradition holds that the chrism itself is nearly 2,000 years old, since some of the remaining chrism is poured into new chrism each time it is made). In the Western church, this has always remained the privilege of the bishop. To this day, when the Catholic Church does a confirmation, the bishop must come to the parish and perform the sacrament, the priests may not. For some reason, the East didn't see a problem with devolving this duty down to the priests, and so now priest chrismate, but of course still use the chrism which is maintained and handed out by the bishop.