This isn't really possible with pure grep, at least not generally.
But if your pattern is suitable, you may be able to use grep multiple times within a pipeline to first reduce your line to a known format, and then to extract just the bit you want. (Although tools like cut and sed are far better at this).
Suppose for the sake of argument that your pattern was a bit simpler: [0-9]+_([a-z]+)_ You could extract this like so:
echo $name | grep -Ei '[0-9]+_[a-z]+_' | grep -oEi '[a-z]+'
The first grep would remove any lines that didn't match your overall patern, the second grep (which has --only-matching specified) would display the alpha portion of the name. This only works because the pattern is suitable: "alpha portion" is specific enough to pull out what you want.
(Aside: Personally I'd use grep + cut to achieve what you are after: echo $name | grep {pattern} | cut -d _ -f 2. This gets cut to parse the line into fields by splitting on the delimiter _, and returns just field 2 (field numbers start at 1)).
Unix philosophy is to have tools which do one thing, and do it well, and combine them to achieve non-trivial tasks, so I'd argue that grep + sed etc is a more Unixy way of doing things :-)