Hello there, I have been facing the same issue, having bought kaspersky premium.
The issue seems to be on kaspersky's side as I have an android which also fails to sync with KPM which makes me believe this problem isn't device specific.
Trouble shooting I have done:
1) Uninstalling and reinstalling KPM, even disconnecting and reconnecting with My Kaspersky account.
2) Using kavremover to remove KPM and reinstalling it again.
I don't use any proxies so I have tried enabling and disabling it with no avail.
If kaspersky had an issue connecting to my account, it just wouldn't show up my vault in my kaspersky account which doesn't seem to be the case as I can access my vault through my kaspersky.
I have tried reinstalling KPM on my phone many times again with no avail.
so according to me the issue relies on kaspersky's side (Maybe Kaspersky premium specific).
Kaspersky support has been of no avail as they just ask me to send my device data and reply "We will act further" so please don't recommend that.
Please find a fix to it as soon as possible as if sync isn't available cross device, there is no use of using KPM.
Thank you