Starting up a 2nd Home assistant server at my vacation home, . Should I use same sign on credentials as I do on my primary home?
You can and there would be no issue.
I would not since password managers (like bitwarden) make this easy.
The Android app allows a three-finger swipe to switch between multiple HASS instances, so it is best to use a different username / hostname. If you set multiple instances as the same, a phone apps may get confused as to which instance of HASS is available.
I’ve also setup remote sites with different hostnames and IPv4 addresses so it is explicit which instance is available (also useful if you intend to use a VPN like TailScale / WireGuard./ NabuCasa for remote access).
It is never good practice to share a password across devices (e.g. were your second home were burgled and the server stolen).
If this helps, this post!