Can I upgrade my rims/tires from 15 inch to 16 inch? Is this possible?

Will this affect performance/handling? Will it require any other adjusts?

2004 Honda Civic
P195/60R15

1997 Pontiac Bonneville
P225/60R16

I want to put the 16s on the Honda.

You want to use the Pontiac rims/tires on the Honda?

If the Pontiac rims have the same lug pattern, they might fit. If the offset of the Pontiac rims is different, that could cause problems.

You can put16s on, but I think those tires are too big

I don't know if the stud pattern of the pontiac is the same as the honda, but those tyres would be way too big for the honda, 225/45R16 will bring the rolling diameter the same

The 16s will have a far bigger diameter than the 15s which will mean -
The tyre is likely to foul something either all the time r when steering or when the suspension compresses over a bump.
The car will be higher off the ground thus making handling poorer
The overall gearing will be increased making it harder to start from a stop and maybe making top gear unusable
The speedometer will read far slower than it should

It is unlikely anyway that the hole pattern for the fixings will match thus making it impossible to fit.
The off set of the wheel may be so different that it fouls the brakes and can't be fitted at all.
Te off set might be so far out that the wheels and tyres protrude too far out from the cars body

195/60R15 tire is a 24 1/8 inch tall tire. The 225/60R16 tire is a 26 5/8 inch tall tire. A difference of 11%. The speedometer will be grossly inaccurate, acceleration will be horrible, handling will be adversely affected because the car isn't designed to have such a large tire AND WORST OF ALL, braking will be so severely limited that you'll hit someone in the rear end. Your brakes can't stop that large a tire. They weren't designed to. 11% bigger means 44% longer stopping distance. You'll dent the front end of your Honda.

And that is IF the larger Pontiac rims will even FIT on your Honda.

Changing wheel and tire size is possible and the whole point is to change performance and handling.

Putting your Pontiac wheels and tires on the tiny Honda is however a total non-starter.

You certainly could use a 16 inch tire and wheel combo on your 2004 Civic. However, to do so you would need to use 16x7 rims with 4-100 bolt pattern.

The Pontiac has a completely different wheel bolt pattern than the Civic. The Bonneville has a 5-115 wheel bolt pattern. It is physically impossible to mount a 5-115 bolt pattern wheel on a vehicle which requires a 4-100 bolt pattern. Get it? The two cars don't even have the same number of bolts!