WinSCP console does not use a terminal emulation.
Consequently, the server shell may use a different set of startup scripts, potentially setting PATH (and other environment variables) differently for WinSCP session than for regular interactive SSH terminals. That may explain the difference. It's generally a misconfiguration, if PATH is set differently for interactive and non-interactive sessions.

I am facing strange issue regarding node.js on a server I am using.

When I log in on that server using typical SSH client like putty or ubuntu on windows that I am using now, this command:
displays, that I am using 10th version of node.

When I will log in through WinSCP and open a terminal or run bash script through execute context menu, I will be using 6th version of node.

I'd like to find out what's going on, why while I am logging in with the same credentials, I am getting different results. Is there a way to configure WinSCP in some way, that I will end up using the "right" version of node (i.e. I'll be in the same environment that I am in when using typical SSH client?).