“...how they knew it was a Catholic priest...”
A Catholic priest would have had the anointing healing oil that is blessed on Holy Saturday - I’m not sure, but don’t think other denominations use this oil.
Full text: Pope Francis’ Chrism Mass homily (historical reflection on OT)
Pope: Homily for Chrism Mass [full text]
Chrism Mass: Thousands celebrate hope and promise [Catholic Caucus]
RITES OF THE BLESSING OF OILS [Catholic/Orthodox Caucus]
THE HOLY OILS [Catholic/Orthodox Caucus]
Chrism Mass: Sharing in the Anointing of the Christ
Why Only One Chrism Mass and Commemorating Saints in Lent
Official text of Pope Benedict XVI's Homily for the Chrism Mass
Priests: Ordinary Men Made Extraordinary by Grace [Holy Orders/Chrism Mass]
Chrism Mass oils have power, Bishop says
Being Catholic: Sacred Things, Holy Oils
The Chrism Mass
Other types of ministers use anointing oil in the prayer for the sick. It’s not just a Catholic “thing”.
Yes, it is a tradition to have the holy oils blessed at a special mass for the priests during Holy Week and often presented at the parishes either at the Holy Thursday mass or on Holy Saturday at the special vigil service/mass for Easter.