Are these two lines the same? How can I say that the two lines are the same: 
Line1=(5,3)+t(-4,8)
Line2=(5,3)+t(-8,16)
They both have the same starting point but can I say they have the same direction vector?  It seems like to me they contain different points for a given "t".

The vector form of the equation you are using is $$\tag{1}{\bf r}(t) = \color{maroon}{\bf p} +t\color{darkgreen} {\bf d}$$
where $\color{maroon}{\bf p}$ gives the initial point on the line (the tip of $\color{maroon}{\bf p}$ is the initial point on the line corresponding to $t=0$) and $\color{darkgreen}{\bf d}$ is the direction vector.
To see why (1) actually gives a line: Note that multiplication of $\color{darkgreen}{\bf d}$ by $t$ just extends, shortens, or reflects $\color{darkgreen}{\bf d}$, it does not change its direction (in the diagram $\color{maroon}{t{\bf d}}$
is the dashed, maroon vector).
So, thinking of adding vectors "tip to tail", the tip of the vector $\color{orange}{{\bf p}+t{\bf d}}$  will always fall on a line.
Note that in (1), we could have used any other vector, such as the blue vector below, that is parallel to $\color{darkgreen}{\bf d}$ as the direction vector,
and the corresponding equation would give the same line, just traced out in a different way.

If you think of the line as being generated by a moving point (the gray one above) whose position at time $t$ is ${\bf r}(t)$, then in your example, the point is moving twice as fast in your   equation for line2 than as in the equation for line1.
