I visited a faux-Irish bar this evening where both three-leaved and four-leaved shamrocks were found in abundance.
The reason why shamrocks are associated with Ireland is that our patron saint, St. Patrick, used the three-leaved plant as a metaphor for the Divine Trinity, the three gods in one, God the Father, God the Son, and God the Holy Ghost. (The Irish usually refer to the Holy Spirit as the Holy Ghost. I spent ages 7–18 at St. Mary’s College CSSp in Dublin, which is run by the Holy Ghost Fathers.)
A four-leaf clover is considered lucky because it is rare. It may be lucky but it is not a shamrock.