Hi there
I haven't done this for a client yet but expect that will occur at some time so pleased you asked. My preference would be to ask your client if he minds that I introduce myself as his personal VA. For that specific moment in time you are 100% at your clients service and I think it's important to acknowledge that in your transactions (verbal, written and financial).
Asking him too means you've covered your base in terms of making sure you know what the client expects but good too to have some preferences of your own handy in case his preference isn't nearly close to yours and thus you might need to do some soft negotiating.
Likely that you will spark their interests on your clients behalf because they will recognise his innovation to use a VA as well, and... it's likely too you are going to get asked what a VA is... which is a great opportunity to market yourself. There's a potential for a great win-win there I think
Like I said though, I haven't had that situation arise yet so am only suggesting what I would do if I were in that position. Hope it's helpful all the same.