Answer:
Apart from the fact they share the same Regnal Name, the two women share a strong sense of duty to God and to their subjects. Both are Protestants, both ascended the throne in their 20's after a period of great controversy.
Elizabeth 1 was accused of treason against her sister Mary 1. She was born into a world where the King, her father, relinquished Papal authority, divorced his wife, and married Elizabeth's mother (Anne Boleyn).
Elizabeth II witnessed first hand a time that can be best described as the biggest constitutional crisis in the history of Britain - and once again dealing with marriage - the abdication of her uncle Edward VIII to marry American divorcee Wallis Simpson thereby putting her father on the throne and making her Heir Presumptive.
Both Elizabeth's witnessed times of war that affected them personally. There are so many other parallels one could draw from the lives of these two Monarchs