Disclaimer: I am not a professional electrician.
The code requirements have changed a bit over the years, and I think they're also a little different for apartments vs. single-family. Having said that, here's what I would expect to have in your situation:
1) One dedicated 20 amp circuit for each bathroom, because hair dryers. The receptacle should be a GFCI.
2) At least one dedicated 20 amp circuit for each kitchen, serving the countertop receptacles, because toasters, coffee pots, waffle irons, etc. It's OK if this circuit serves more than one receptacle in the kitchen, but it shouldn't serve anything in any other room. The first receptacle in the string should be a GFCI. (For a single-family residence, you need two dedicated 20 amp circuits for each kitchen.)
3) At least one 15 amp circuit for the lights and general-purpose outlets in each unit. Depending on how many lights and outlets each unit has, maybe two 15 amp circuits. These don't have to have GFCIs, but if your city requires you to come up to the very latest code, the receptacles might have to have a thing called an AFCI. (Editorial: The electrical manufacturers realized they made a mistake when making GFCIs available as a $60 circuit breaker or a $7 receptacle - nobody bought the breakers. So they wrote AFCIs into the electrical code, and only made them available, so far, as $60 circuit breakers.)
If you're wiring brand new, one theory is to put all the lights on one circuit, and all the receptacles on another, so that if somebody plugs something in that pops the receptacle breaker, they aren't also plunged into darkness. This isn't required, though, as far as I know.
To some specific points:
The fridge doesn't have to have its own dedicated circuit, unless it's some monster $10,000 Sub-Zero 30-cubic-foot built-in thing.
The $700 GE 18-cubic-foot just doesn't need that much juice. :) It
can go on one of the "general" lighting and receptacle circuits.
Modern practice would probably be to put the dishwasher (if equipped)
on its own 15-amp circuit - the heater pulls a lot of juice in the dry
cycle - and the DIsposall (if equipped) on its own 15-amp circuit. You
could probably get away with putting both of those on one 15-amp or
20-amp circuit of their own. There are also plenty of houses where the dishwasher and Disposall run off of one of the 20-amp kitchen circuits or one of the 15-amp general lighting circuits, and it works just fine.
> I go into her room and she has 2 fans going, an alarm clock, an Iphone charger, all hooked up to the same power strip and (the kicker) her hair dryer.
The hair dryer is what's doing it. The fans are maybe 80 watts each. The alarm clock is 5 to 10 watts. The iPhone charger is 10 watts. Total maybe 180 watts. A typical 15 amp circuit is good for 1,800 watts, or 10 times that. The "high" setting of most hair dryers is designed to be 1,800 watts - the full capacity of a 15 amp circuit.
That panel looks to be relatively recent, and isn't one of the "bad" brands (Federal Pacific Eletric, Pushmatic, Zinsco), so you shouldn't need to replace the whole panel. If it needs another breaker, they are readily available at the hardware store.
From the tenant's description , it does sound like there is too much stuff on that one circuit. You will never be able to get it to balance out with exactly 10 lights/receptacles on one circuit and 10 lights/receptacles on the other, but if you can get it closer to even, that would be good.
I agree with the other posters that you probably want an electrician to look at the unit now.
You might check with the city/county to see what they require when making renovations. Sometimes they require that if you touch anything, the whole unit has to be brought up to 2019 code; sometimes only the stuff you touch has to be brought up to 2019 code; sometimes it only has to be brought up to some earlier year code. For houses built from maybe the 1960s on up, they sometimes just ask it to meet code for the year it was built, but that won't apply to an 1897 house. The reason you should check this is that sometimes the electrician will say you need to replace the outlets, switches, fixtures, wiring, breaker panel, meter socket, meter, the transformer outside, and the power plant across town, but maybe you don't actually have to do that much. :)
If they are still running incandescent lamps, you can buy a little headroom quickly by switching to LED. Swapping two 60-watt incandescents to LEDs will cut the power for that fixture from 120 watts to about 18 watts. If you do this, make sure you get the "warm white", 2700 K or 2750 K LEDs (it will say on the box), because the color on those is most similar to incandescents. Most people find the 3000 K or higher LEDs "too blue" and odd-looking. If the lamps are already LED or compact fluorescent, this won't help much.
One thing you might ask the electrician to check is if any of the outlets are "back-stabbed", "stabbed", or "back-wired". Some outlets have a hole in the back, so you can just strip the wire and shove it straight into the hole, where a small spring clip retains it. (Go look at a new outlet at the hardware store to see.) The spring tends to loosen up over the years, making the connection heat up. This creates more voltage drop - lower voltage at the outlets past that one - and can start a fire if it's bad enough. The right way is to loop the wire around the screws on the side of the outlet and tighten the screw - if you have copper wire, and you get the screw tight enough the first time, it will stay that way for 50+ years. This won't immediately fix your breaker tripping problem, but it's something you should ask them to check on any of the outlets they touch.
Something you can check yourself is if all the outlets are gripping their plugs tightly. The easiest way to do this is with a 2-prong plug for anything, even your cell phone charger. Plug it in to a new outlet at the hardware store to see what it should feel like, then try the same plug in all outlets at the unit. They may not all be quite as strong as the new one, but sometimes you find one that is really loose - if so, replace that outlet, or have it replaced. Again, this won't fix your breaker-tripping problem directly, but it will help a little.
In the future, like if one of your apartments turns over, a handy thing to have is a "Kill-a-watt" plug-in watt-hour meter (about $30 to $40 at the hardware store, electronics store, or Amazon) and a space heater or hair dryer ($20). Ideal and Fluke make dedicated tools for this, but they are $300+ .
Go to an outlet, plug in your meter, and write down the reading. It's normal for it not to be exactly 120.0 V AC, and it's normal for it to bounce around a few tenths of a volt while you watch it. If it's consistently way low (below maybe 110.0 V) or way high (above maybe 125.0 V), you may want to seek help from an electrician before proceeding. Now, plug in the space heater or hair dryer, and crank it up to high. Watch the voltmeter while you do this. It is totally normal for the voltage to drop down when the heater/dryer kicks in, but let the heater/dryer run for a couple of minutes and write down the lowest voltage you see. Then, turn the heater/dryer off and do some math.
If the voltage dropped 5% or less (like, it was 120.0 V with the heater off, and 114.0 V at its lowest with the heater on), then the circuit from the breaker panel to that outlet is in pretty good shape. If it's between 5% and maybe 8%, that's not as good, but it might be OK. If it's over 10%, then you should investigate further, or have an electrician investigate - you've got a loose connection somewhere between that outlet and the breaker box, which is potentially dissipating a lot of heat inside the wall somewhere. Back-stabbed connections, loose wire nuts, and terminal screws that aren't tightened all the way down on receptacles and switches can cause this problem.
I hope this helps!