I simply select the profiling icon in the dbg toolbar and press f5.

Please note that the same procedure without openvpn works. I use phped for 15 years
This is just the first time I use phped ssh tunnel + openvpn

Dbg toolbar in browser I guess? Can you try that in the IDE by clicking profiler icon in the IDE toolbar?
As of browser toolbar, obviously it wasn't configured to instruct the debugger to use ssh tunnel. What I noticed is "clienthost" in the error message. IT should be either localhost or 127.0.0.1 -- they are the tunnel connection points for debugger to connect to.

good guess! yes, Dbg toolbar in the browser. I've tried changing clienthost with localhost and 127.0.0.1, but none works

I still see
Reason: failed to connect to the client at <vpn ip>:7869, please make sure that IDE is running

but if I run the profiler in PHPED it works!

Do you have any idea on what setting I should put in the DBG browser options to make it working from there?

Extra (OT) question: is there a reason that SQL and Memory Profiler works only if profiler was started in PHPED and not in the browser?

In IDE it works because IDE knows about your project. As of toolbar -- it has no clue about it.
If you're working with this toolbar https://chrome.google.com/webstore/detail/dbg-debugger-toolbar/jpfjdjkohmopcgecijakgemceodpjpdp?hl=en-US
just enter localhost as "development machine".

Yes I'm working with that toolbar, but local host is not working for me.
Should I restart the browser after changing this setting?

Is it normal that sql profiler never works from browser toolbar?