What command are you using?
Can you run scp and increase the verbosity ("scp -v ...")? That will show you more info that will help you debug.
From my linux machine "scp file.txt admin@merlin:/tmp/mnt/storage" executed error-free. Like you, my Asus device is not running ftp, samba, nfs, etc.
Thanks for looking at this, I did just increase the verbosity, and here is what I got (edited out actual IPs and addresses):
root@myserver:~# scp -v /var/www/html/index.php myaccount@myrouteraddress:/tmp/mnt/Backup
Executing: program /usr/bin/ssh host myrouteraddress, user myaccount, command scp -v -t /tmp/mnt/Backup
OpenSSH_6.6.1, OpenSSL 1.0.1f 6 Jan 2014
debug1: Reading configuration data /etc/ssh/ssh_config
debug1: /etc/ssh/ssh_config line 19: Applying options for *
debug1: Connecting to myrouteraddress [xxx.xxx.xxx.xxx] port 22.
debug1: Connection established.
debug1: permanently_set_uid: 0/0
debug1: identity file /root/.ssh/id_rsa type 1
debug1: identity file /root/.ssh/id_rsa-cert type -1
debug1: identity file /root/.ssh/id_dsa type -1
debug1: identity file /root/.ssh/id_dsa-cert type -1
debug1: identity file /root/.ssh/id_ecdsa type -1
debug1: identity file /root/.ssh/id_ecdsa-cert type -1
debug1: identity file /root/.ssh/id_ed25519 type -1
debug1: identity file /root/.ssh/id_ed25519-cert type -1
debug1: Enabling compatibility mode for protocol 2.0
debug1: Local version string SSH-2.0-OpenSSH_6.6.1p1 Ubuntu-2ubuntu2.6
debug1: Remote protocol version 2.0, remote software version dropbear_2014.66
debug1: no match: dropbear_2014.66
debug1: SSH2_MSG_KEXINIT sent
debug1: SSH2_MSG_KEXINIT received
debug1: kex: server->client aes128-ctr hmac-md5 none
debug1: kex: client->server aes128-ctr hmac-md5 none
debug1: sending SSH2_MSG_KEX_ECDH_INIT
debug1: expecting SSH2_MSG_KEX_ECDH_REPLY
debug1: Server host key: ECDSA xx:2f:xx:xx:xx:e7:f9:0e:e0:1c:xx:d4:cf:fc:xx:xx
debug1: Host 'myserver' is known and matches the ECDSA host key.
debug1: Found key in /root/.ssh/known_hosts:1
debug1: ssh_ecdsa_verify: signature correct
debug1: SSH2_MSG_NEWKEYS sent
debug1: expecting SSH2_MSG_NEWKEYS
debug1: SSH2_MSG_NEWKEYS received
debug1: SSH2_MSG_SERVICE_REQUEST sent
debug1: SSH2_MSG_SERVICE_ACCEPT received
debug1: Authentications that can continue: publickey
debug1: Next authentication method: publickey
debug1: Offering RSA public key: /root/.ssh/id_rsa
debug1: Server accepts key: pkalg ssh-rsa blen 279
debug1: key_parse_private2: missing begin marker
debug1: read PEM private key done: type RSA
debug1: Authentication succeeded (publickey).
Authenticated to myrouteraddress ([xxx.xxx.xxx.xxx]:22).
debug1: channel 0: new [client-session]
debug1: Entering interactive session.
debug1: Sending environment.
debug1: Sending env LANG = en_US.UTF-8
debug1: Sending command: scp -v -t /tmp/mnt/Backup
Sending file modes: C0644 418 index.php
Sink: C0644 418 index.php
scp: /tmp/mnt/Backup/index.php: Invalid argument
root@myserver:~# debug1: client_input_channel_req: channel 0 rtype exit-status reply 0
debug1: channel 0: free: client-session, nchannels 1
debug1: fd 0 clearing O_NONBLOCK
debug1: fd 1 clearing O_NONBLOCK
Transferred: sent 3376, received 1720 bytes, in 0.4 seconds
Bytes per second: sent 8429.0, received 4294.4
debug1: Exit status 1
...and then, nothing is actually transferred to the USB disk attached to the router