Simultaneous voice and cellular data on the iPhone 6S Plus on Verizon
You are right when you state that simultaneous data and voice is possible over the LTE connection. There's not a VoLTE-capable network in the world where this isn't the case.

But many of your other assumptions, albeit founded on sound logic, are not consistent with reality.

So here's what you need to know:


Your calls are definitely not occurring over LTE if the data signal drops when you place the call. But your phone is definitely capable of VoLTE (as is Verizon's network).
Your Android phone has two radios in it, which is why you can maintain your LTE data connection while on a phone call on that phone. The vast majority of Verizon-sanctioned LTE-capable Android phones have two radios in them (although we're starting to see some hit the market with only a single radio now that VoLTE is possible). The phone call goes over 1xRTT while the LTE radio maintains data connectivity.


There are several things to check before opening a trouble ticket with Verizon:


Make sure "Voice &amp; Data" is selected in Settings > Cellular > Enable LTE.
Make sure your Verizon account has the feature called "Advanced Calling 1.0" added to it. You can add this through My Verizon yourself (in the same place where you'd activate/deactive the mobile hotspot feature).
Make sure you are using a newer SIM card. Believe it or not, there are old Nano-SIM cards out there that simply will not support VoLTE. If you need a new one, ask Verizon.


If you've done all of this, there's one more thing to know before opening a trouble ticket: your phone will still prefer 1xRTT phone calls over LTE ones if you're in a marginal service area. And when it does this, since no iPhone has ever supported TX/RX with two cell networks simultaneously, your iPhone will drop LTE and hop straight onto 1xRTT to grab the call.

If marginal service is the cause of your problem, then your only solution will be to use a different phone—one that has a dual radio setup. Manufacturers are rapidly moving away from dual radios though, so be careful when purchasing anything new.

Good luck!