Backup SQL Server with VMware
I would prefer not to use this backup solution but I have no choice
  but to cope with it.


You are thinking absolutely correct. Please note that VMware backups take backup using snapshot mechanism its not like backup you take using TSQL Command. Its not a good idea to rely on VM backups I strongly suggest you to start takign TSQL backup specially if database recovery model is full and you want to take transaction log backup to truncate logs


  SQL Server thinks it has performed a full backup of the databases to a virtual device (physical_device_name = GUID).


This is how SQl Server thinks with any other third party tool. If you use any othet third party tool to take backup the VIRTUAL_DEVICE would be the location where you took backup and this would be logged into errorlog.


  Do you know why it works this way? Is VMware aware of SQL Server existence? Or maybe SQL Server has wrong data in msdb?


To explain how it works you must read Information Shedding Light on VDI VSS backups


  Can I run into any problems with that solution (other than trying to restore the DB from the virtual device)?


These are just snapshot backups which are not useful when you want to do point int time recovery.