Usually paste prints two named (or equivalent) files in adjacent columns like this:

paste <(printf '%sn' a b) <(seq 2)

Output:

a 1
b 2

But when the two files are /dev/stdin and /dev/stderr, it doesn't seem to work the same way:

seq 2 | tee >(sed 's/^/e/' > /dev/stderr) | paste /dev/stdin /dev/stderr

Output:

1 e1
e2
^C

It hangs -- ^C means pressing Control-C to quit.

Changing the | to a ; also doesn't work:

seq 2 | tee >(sed 's/^/e/' > /dev/stderr) ; paste /dev/stdin /dev/stderr

Output:

1
2
e1
e2
^C

Also hangs -- ^C means pressing Control-C to quit.

Desired output:

1 e1
2 e2

Can it be done using paste? If not, why not?

Why you can't use /dev/stderr as a pipeline

The problem isn't with paste, it's with the available file descriptors. All commands have one input descriptor, /dev/stdin, and two outputs, /dev/stdout and /dev/stderr.

When you run

seq 2 | tee >(sed ... >/dev/stderr)

with nothing after it, the /dev/stdout of tee is the console -- a write-only file descriptor, where command output usually appears. The output of sed has been redirected to /dev/stderr, which points to that same console (because you haven't redirected it with eg. 2>...). When you add another command to the pipeline...

seq 2 | tee >(sed ... >/dev/stderr) | paste /dev/stdin /dev/stderr

the second | connects the /dev/stdout of tee to /dev/stdin of paste. But there's no equivalent read descriptor to connect up to /dev/stderr/. sed output still goes to the console.
Meanwhile, when paste can reads its first argument (/dev/stdin) fine, but when it attempts to read from the second filename, /dev/stderr, that's the console, which cannot be read and hangs.

Your link Why can't I read /dev/stdout with a text editor? is still relevant here because those same restrictions apply to /dev/stderr.

How to make a second pipeline

It appears you want to use tee to "split" some output, process one of the copies with an additional command, and paste the two back together. That means two concurrent pipes, one for each column. The shell pipeline ... | ... provides one of those. bash has created another for the >(...), but that's an anonymous FIFO -- you can't retrieve its name for use in another command. So you're going to need to create another, using a name you do know:

mkfifo RHS
seq 2 | tee >(sed 's/^/e/' > RHS ) | paste /dev/stdin RHS

If this is for use in a script, you may need to make that FIFO in a temporary directory, and remove it after use. bash will automatically clear up the one it created.