Poor Katherine Grey. She didn’t have a lot of choice in the way her life went in My Lady Jane. She was married twice in the series, and neither of those marriages she necessarily wanted.
The first was a definite turn-off. She was forced to marry Lord Leicester. And no, this didn’t happen in real life. Katherine’s first marriage was to Henry Herbert, 2nd Earl of Pembroke, and that marriage was annulled by Henry. Her second marriage was to a certain Lord Seymour.
My Lady Jane didn’t disclose which Lord Seymour we met
There were multiple Lord Seymours in real history. I suspected that Dominic Cooper’s Lord Seymour was Edward Seymour, 1st Duke of Somerset. This would make sense with the time period, as Edward was Lord Protector up until 1549. This would also make sense for the storyline of Katherine marrying his son. There's just a slight problem with this as Edward Seymour was executed for treason in 1552. My Lady Jane seems to take place in 1553.
Edward Seymour had a son, also called Edward. This could make more sense as it seemed that the show took place in 1553. Despite his father's execution, Queen Mary and later Queen Elizabeth stepped in to restore some wealth and give Edward the title of Earl of Hertford.
If Cooper’s character was the elder Edward Seymour, then it does connect to Katherine Grey. Katherine did marry Edward's son, Edward, Early of Hertford. However, the marriage wasn’t something Mary had arranged. Katherine was viewed as the likely successor to the throne after Queen Elizabeth I, who was known as The Virgin Queen. Elizabeth was angry when Katherine secretly married Edward Seymour. There was an issue as Edward gave Katherine a document that proved their marriage, but Katherine lost it.
Katherine spent her final years imprisoned in the Tower of London. The marriage to Edward was annulled, and their children were considered illegitimate.
So, it is possible that the marriage of Katherine and Lord Seymour in My Lady Jane was the one from real life. The real life marriage was under some very different circumstances.