6 years is the length of the term of a US Senator. A person elected to the US Senate is called a US Senator. A "state senator" is one elected to the upper chamber of a state legislature. In my home state of Washington, the term for a Senator is 4 years. The length of the term varies by state.
A US Senator is elected for 6 years. They are divided into three groups, so that one third are up for election every two years. And only one state senator is up for election at any election cycle (unless there is a mid-term election to replace a senator who has died or resigned).
Lincoln was never a US senator. He served one term -- two years -- in Congress long before he was President. He ran for Senator in 1858, attracted national attention for his debates with Stephen Douglas, but lost the election to Douglas.