Yep, replicated with debugging enabled, thanks Igor

Looks like the installer is trying to enable the InnoDB engine when backing up / restoring the databases and is looking for "have_innodb" variable which doesn't exist in MariaDB 10.1 and 10.2. Unfortunately I haven't got a 10.0 instance to test against.
There you go:
[...]
+ /usr/bin/mysqldump --user=admin --quote-names --add-drop-database --databases mysql psa apsc horde
+ rc=0
+ unset MYSQL_PWD
+ return 0
+ suc
+ p_echo done
+ '[' -n 1 -o -n '' -o -z /var/log/plesk/install/plesk_17.5.3_installation.log ']'
+ echo done
+ p_echo ' MySQL databases are dumped to /var/lib/psa/dumps/mysql.preupgrade.17.5.3-17.5.3.20180227-091316.dump.gz'
+ '[' -n 1 -o -n '' -o -z /var/log/plesk/install/plesk_17.5.3_installation.log ']'
+ echo ' MySQL databases are dumped to /var/lib/psa/dumps/mysql.preupgrade.17.5.3-17.5.3.20180227-091316.dump.gz'
+ unset MYSQLDUMP_OPTIONS
+ SAFE_PREP_DB_BACKUP=/var/lib/psa/dumps/mysql.preupgrade.17.5.3-17.5.3.20180227-091316.dump.gz
+ transaction_add_rollback_action safe_prep_db_upgrade_rollback
+ '[' -z true ']'
+ '[' -z 'core_start_product_on_rollback base_rollback_product_config_default base_rollback_product_config show_report_after_rollback unlock_bootstrapper' ']'
+ TRANSACTION_ROLLBACK_FUNCS='safe_prep_db_upgrade_rollback core_start_product_on_rollback base_rollback_product_config_default base_rollback_product_config show_report_after_rollback unlock_bootstrapper'
+ enable_innodb_engine
+ local have_innodb=
+ '[' -z '' ']'
+ local mysql_db_name=mysql
+ local db_select_output=
+ db_select 'SHOW VARIABLES LIKE '\''have_innodb'\'''
+ local 'desc=execute SQL query'
+ local 'query=SHOW VARIABLES LIKE '\''have_innodb'\'''
++ mysql_raw -e 'SHOW VARIABLES LIKE '\''have_innodb'\'''
+ local output=
+ local status=0
+ '[' 0 -ne 0 ']'
+ db_select_output=
+ return 0
++ get_narg 2
++ shift 2
++ return 0
+ have_innodb=
+ '[' -n '' ']'
+ have_innodb=NO
+ echo_try 'allow INNODB engine'
+ msg='allow INNODB engine'
+ pnnl_echo ' Trying to allow INNODB engine... '
+ '[' -n 1 -o -n '' -o -z /var/log/plesk/install/plesk_17.5.3_installation.log ']'
+ echo -n ' Trying to allow INNODB engine... '
+ find_my_cnf
+ local non_fatal=
+ local 'cnf_files=/etc/my.cnf /etc/mysql/my.cnf /var/db/mysql/my.cnf'
+ for my_cnf in '$cnf_files'
+ '[' -f /etc/my.cnf ']'
+ break
+ '[' -f /etc/my.cnf -o -n '' ']'
+ '[' -f /etc/my.cnf ']'
+ grep -q '^skip-innodb' /etc/my.cnf
+ suc
+ p_echo done
+ '[' -n 1 -o -n '' -o -z /var/log/plesk/install/plesk_17.5.3_installation.log ']'
+ echo done
+ pnnl_echo 'Restarting mysql'
+ '[' -n 1 -o -n '' -o -z /var/log/plesk/install/plesk_17.5.3_installation.log ']'
+ echo -n 'Restarting mysql'
+ '[' 2 -le 2 ']'
+ local service_name=mysql
+ local action=restart
+ local ret=0
+ local inten
+ shift
+ shift
+ test linux = linux
+ is_function mysql_restart_linux_redhat
++ type -t mysql_restart_linux_redhat
+ local type_output=
+ test X = Xfunction
+ is_function mysql_restart_linux
++ type -t mysql_restart_linux
+ local type_output=
+ test X = Xfunction
+ is_function mysql_restart
++ type -t mysql_restart
+ local type_output=
+ test X = Xfunction
+ eval 'service=$mysql_service'
++ service=mysql
+ '[' -n mysql ']'
+ inten='restart service mysql'
+ '[' restart = status -o restart = exists ']'
+ echo_try 'restart service mysql'
+ msg='restart service mysql'
+ pnnl_echo ' Trying to restart service mysql... '
+ '[' -n 1 -o -n '' -o -z /var/log/plesk/install/plesk_17.5.3_installation.log ']'
+ echo -n ' Trying to restart service mysql... '
+ local action=restart
+ local service=mysql
+ local service_name=mysql
+ '[' restart '!=' exists ']'
+ _service_exec mysql exists
+ local service=mysql
+ local action=exists
+ local action_cmd
+ local sysvinit_service=/etc/init.d/mysql
+ '[' -x /bin/systemctl ']'
+ case "${action}" in
+ /bin/systemctl list-unit-files
+ awk 'BEGIN { rc = 1 } $1 == "mysql.service" { rc = 0;} END { exit rc }'
+ return 0
+ '[' 0 '!=' 0 ']'
+ case "$action" in
+ pleskrc mysql status
+ '[' 2 -le 2 ']'
+ local service_name=mysql
+ local action=status
+ local ret=0
+ local inten
+ shift
+ shift
+ test linux = linux
+ is_function mysql_status_linux_redhat
++ type -t mysql_status_linux_redhat
+ local type_output=
+ test X = Xfunction
+ is_function mysql_status_linux
++ type -t mysql_status_linux
+ local type_output=
+ test X = Xfunction
+ is_function mysql_status
++ type -t mysql_status
+ local type_output=function
+ test Xfunction = Xfunction
+ mysql_status
+ local file
+ '[' Xredhat = Xdebian ']'
+ '[' -z yes ']'
+ '[' yes = yes ']'
+ service_ctl status mysql mysql
+ local action=status
+ local service=mysql
+ local service_name=mysql
+ '[' status '!=' exists ']'
+ _service_exec mysql exists
+ local service=mysql
+ local action=exists
+ local action_cmd
+ local sysvinit_service=/etc/init.d/mysql
+ '[' -x /bin/systemctl ']'
+ case "${action}" in
+ /bin/systemctl list-unit-files
+ awk 'BEGIN { rc = 1 } $1 == "mysql.service" { rc = 0;} END { exit rc }'
+ return 0
+ '[' 0 '!=' 0 ']'
+ case "$action" in
+ _service_exec mysql status
+ local service=mysql
+ local action=status
+ local action_cmd
+ local sysvinit_service=/etc/init.d/mysql
+ '[' -x /bin/systemctl ']'
+ case "${action}" in
+ action=is-active
+ /bin/systemctl is-active mysql.service
+ return 0
+ return 0
+ _service_exec mysql restart
+ local service=mysql
+ local action=restart
+ local action_cmd
+ local sysvinit_service=/etc/init.d/mysql
+ '[' -x /bin/systemctl ']'
+ case "${action}" in
+ /bin/systemctl restart mysql.service
+ ret=0
+ '[' restart '!=' status -a restart '!=' exists ']'
+ '[' 0 -eq 0 ']'
+ suc
+ p_echo done
[...]