There are 3 issues here. The one in the title is actually the least of the concerns:
The US electric supply has gradually increased the standard voltage over the past century from 110V to 115V to 120V. Or rather, 2x that - 220V to 230V to 240V, with most small appliances and lighting using half that voltage. So the change from 115V to 120V is of no real consequence, and this change is within the variance that most devices can handle.
The output voltage of a transformer is a ratio of the input voltage. So a 115V/10V transformer is also a 120V/10.5V transformer, which is just fine. But a 120V/16V transformer is producing more than 50% higher output voltage and that may be too much for the old doorbell. Or it may be just fine. Unfortunately, finding out whether an old doorbell will work on 16V may not be so easy. It may work fine, it may not work at all, or it may be damaged by use with the new transformer (either immediate or long-term).
This is a tough one. A new transformer will normally tell you the total power. But figuring out the old one may not be so easy. My hunch is that a new 120V/16V 30VA transformer probably has enough power to handle an old doorbell together with the new one. The catch is the Ring doorbell specs are a bit vague "10VA to 40VA".
My recommendation is to install a new transformer for the new doorbell and keep the old transformer for the old doorbell. This should be easy enough to do as long as the splice is accessible so that you can split the wiring of the two doorbells.