Have you noticed that the _ was "ignored" as well? That's a big hint.
If you use set -u, you'll see the following:
-bash: name_: unbound variable
The way bash parses it, the underscore is part of the variable name.
There are several ways to fix the problem.
The cleanest is the ${var} construct which separate the variable name from its surroundings.
You can also use quotation in various ways to force the right parsing, e.g.: "$dir/$name""_$date.ext"
And in case your variables might contain spaces (now, or in the future) use quotation for words.
command >"$dir/${name}_$date.ext"
command >"${dir}/${name}_${date}.ext"
Both these are fine, just pick one style and stick to it.
command > "${dir}/${name}_${date}.ext"set -umight be a good idea.name_. When you cannot change the last line into fields with{}, you can work around withname_=bob_.