SFTP with chroot depending on public key of connecting user

the chroot=... does not work.

No, there is nothing like that in the manual page for sshd, describing the format of authorized_keys file.

If you would put chroot into command=, you would not be able to use internal-sftp, because it is substitution of internal function call inside sshd.

Recommended way is set up more users, if you need separation. You might also use arguments to internal-sftp, if you don't need strict separation (for exaxmple just different working directories), such as

command="internal-sftp -d /backup/client-1/data" ssh-rsa (key1)

There is also possible to limit amount of requests using -P option as in manual page for sftp-server.

Alternatively, are there other SFTP servers that can do what I want?

yes, you can use proftpd

Prepare user environment. With ProFTPD there is no need to give to user a valid shell.

# useradd -m -d /vhosts/backup/user1/ -s /sbin/nologin user1
# passwd --lock user1
Locking password for user user1.
passwd: Success

# mkdir /vhosts/backup/user1/.sftp/
# touch /vhosts/backup/user1/.sftp/authorized_keys

# chown -R user1:user1 /vhosts/backup/user1/
# chmod -R 700 /vhosts/backup/user1/

In order to use OpenSSH public keys in a SFTPAuthorizedUserKeys, you must convert them to the RFC4716 format. You can do this with ssh-keygen tool:

# ssh-keygen -e -f user1.public.key > /vhosts/backup/user1/.sftp/authorized_keys

Setup ProFTPD

ServerName "ProFTPD Default Installation"
ServerType standalone
DefaultServer off

LoadModule mod_tls.c
LoadModule mod_sftp.c
LoadModule mod_rewrite.c

TLSProtocol TLSv1 TLSv1.1 TLSv1.2

# Disable default ftp server
Port 0

UseReverseDNS off
IdentLookups off

# Umask 022 is a good standard umask to prevent new dirs and files
# from being group and world writable.
Umask 022

# PersistentPasswd causes problems with NIS/LDAP.
PersistentPasswd off

MaxInstances 30

# Set the user and group under which the server will run.
User nobody
Group nobody

# Normally, we want files to be overwriteable.
AllowOverwrite                  on

TimesGMT off
SetEnv TZ :/etc/localtime

<VirtualHost sftp.example.net>
    ServerName "SFTP: Backup server."
    DefaultRoot ~
    Umask 002
    Port 2121

    RootRevoke on

    SFTPEngine on
    SFTPLog /var/log/proftpd/sftp.log

    SFTPHostKey /etc/ssh/ssh_host_rsa_key
    SFTPHostKey /etc/ssh/ssh_host_dsa_key
    SFTPDHParamFile /etc/pki/proftpd/dhparam_2048.pem
    SFTPAuthorizedUserKeys file:~/.sftp/authorized_keys

    SFTPCompression delayed
    SFTPAuthMethods publickey
</VirtualHost>

<Global>
    RequireValidShell off
    AllowOverwrite yes

    DenyFilter \*.*/

    <Limit SITE_CHMOD>
        DenyAll
    </Limit>
</Global>

LogFormat default "%h %l %u %t \"%r\" %s %b"
LogFormat auth    "%v [%P] %h %t \"%r\" %s"
ExtendedLog /var/log/proftpd/access.log read,write

Create DH (Diffie-Hellman ) group parameters.

# openssl dhparam -out /etc/pki/proftpd/dhparam_2048.pem 2048

Configure any SFTP client. I have used FileZilla

If you run ProFPTD in debug mode

# proftpd -n -d 3 

In the console you will see something like the following

2016-02-21 22:12:48,275 sftp.example.net proftpd[50511]: using PCRE 7.8 2008-09-05
2016-02-21 22:12:48,279 sftp.example.net proftpd[50511]: mod_sftp/0.9.9: using OpenSSL 1.0.1e-fips 11 Feb 2013
2016-02-21 22:12:48,462 sftp.example.net proftpd[50511] sftp.example.net: set core resource limits for daemon
2016-02-21 22:12:48,462 sftp.example.net proftpd[50511] sftp.example.net: ProFTPD 1.3.5a (maint) (built Sun Feb 21 2016 21:22:00 UTC) standalone mode STARTUP
2016-02-21 22:12:59,780 sftp.example.net proftpd[50512] sftp.example.net (192.168.1.2[192.168.1.2]): mod_cap/1.1: adding CAP_SETUID and CAP_SETGID capabilities
2016-02-21 22:12:59,780 sftp.example.net proftpd[50512] sftp.example.net (192.168.1.2[192.168.1.2]): SSH2 session opened.
2016-02-21 22:12:59,863 sftp.example.net proftpd[50512] sftp.example.net (192.168.1.2[192.168.1.2]): Preparing to chroot to directory '/vhosts/backup/user1'
2016-02-21 22:12:59,863 sftp.example.net proftpd[50512] sftp.example.net (192.168.1.2[192.168.1.2]): Environment successfully chroot()ed
2016-02-21 22:12:59,863 sftp.example.net proftpd[50512] sftp.example.net (192.168.1.2[192.168.1.2]): USER user1: Login successful

And the follwoing lines in a /var/log/sftp.log

2016-02-21 22:12:48,735 mod_sftp/0.9.9[50309]: sending acceptable userauth methods: publickey
2016-02-21 22:12:48,735 mod_sftp/0.9.9[50309]: public key MD5 fingerprint: c2:2f:a3:93:59:5d:e4:38:99:4b:fd:b1:6e:fc:54:6c
2016-02-21 22:12:48,735 mod_sftp/0.9.9[50309]: sending publickey OK
2016-02-21 22:12:59,789 mod_sftp/0.9.9[50309]: public key MD5 fingerprint: c2:2f:a3:93:59:5d:e4:38:99:4b:fd:b1:6e:fc:54:6c
2016-02-21 22:12:59,790 mod_sftp/0.9.9[50309]: sending userauth success
2016-02-21 22:12:59,790 mod_sftp/0.9.9[50309]: user 'user1' authenticated via 'publickey' method

P.S.

The configured path for a file containing authorized keys (SFTPAuthorizedUserKeys) can use the %u variable, which will be interpolated with the name of the user being authenticated. This feature supports having per-user files of authorized keys that reside in a central location, rather than requiring (or allowing) users to manage their own authorized keys. For example:

SFTPAuthorizedUserKeys file:/etc/sftp/authorized_keys/%u

I want several clients to be able to store files on my server. Each client should not be able to see any other client's files. And I do not want to litter my server with dozens of user accounts, so I'd like an easily manageable solution for the clients to share a user account and still have no access to eachother's files.

with ProFTPD it's possible too. You just need a little modify my initial configuration

<VirtualHost sftp.example.net>
    ...   
    SFTPAuthorizedUserKeys file:/etc/proftpd/sftp_authorized_keys
    AuthUserFile /etc/proftpd/sftp_users.passwd

    CreateHome on 0700 dirmode 0700 uid 99 gid 99

    RewriteHome on
    RewriteEngine on
    RewriteLog /var/log/proftpd/rewrite.log
    RewriteCondition %m REWRITE_HOME
    RewriteRule (.*) /vhosts/backup/%u
</VirtualHost>

And create one virtual account

# ftpasswd --passwd --file /etc/proftpd/sftp_users.passwd --sha512 --gid 99 --uid 99 --shell /sbin/nologin --name user1 --home /vhosts/backup

That's all. For every additional account all you need is to add his public key to the /etc/proftpd/sftp_authorized_keys

Note: the file must contain new line in the end! It's important.