She vacated NY-20 after 2008 in order to fill Hillary's Senate seat for the remainder of Hillary's second term, which ended in 2012, when Kirsten ran again, successfully. In 2018, she will be up for her second full senate term.
NY-20 is currently held by Democrat Paul Tonko, who won in 2016 with 64% of the vote.
The NY-20 Gillibrand won and served in is not the same as the post-2013 NY-20. That is a historically Republican upstate district (elected the legendary Gerry Solomon prior to John Sweeney, who lost in an upset to Gillibrand in the ‘06 anti-GOP landslide). The seat was reclaimed in 2010 by the GOP and John Faso sits in the seat today, which is now the 19th.
The present 20th is based in Albany, which was the 21st district prior to 2013. That is a historically Dem district that hasn’t elected a Republican since 1968.
I noticed that too. I wonder if Paul is amused to discover that he's just a placeholder in Kirsten's seat. I suppose the underlying idea is that "once a woman's seat, always a woman's seat."