Are the inside parts of toilets universal?

No. A lot of them work with each other, but not all toilets are the same.

Sort of

For most toilets there's universal fittings that will work just fine, you may need to adjust them a little bit, but they're made to be adjusted, and they'll work just fine with most toilets.

If you have the original factory parts in your toilet, they may not be adjustable, and if you tried to swap them into another toilet they may not fit/work in other brands/models, or they may kind of work, but maybe not quite right.

There are a handful of brands that don't tend to play well with the universal fit parts, I want to say Kohler is one, and if you go to a hardware store, most likely they're going to stock the universal parts, then a couple of the most common oddball brands.

There's also of course some weird toilets that are just totally different- pressure assisted flush, composting or incinerator toilets, etc. that aren't even working on the same principle as most toilets, but I think the odds are that if you have one of those, you know that already.

Also I haven't played with any toilets that were manufactured that way, but I did retrofit one of my toilets to be a dual-flush. Those kits seem pretty universal, but probably double-check before trying to put them in an oddball toilet.

They are not all the same.

Measure the diameter of the hole at the bottom of the water holding tank. It's the main difference between older and newer toilets in the US.

Any US toilet repair kit should list what diameter(s) it supports.

Depth of the holding tank will vary as well, but most repair kits account for this. Some kits may require using a hand saw to cut some plastic tubes to fit smaller tanks. Other kits have an extendable or collapsible tube.

After trying to make the wrong replacement handle work, I'm gonna cast my vote as no.

No, but most replacement parts are semi universal parts that can be adjusted to fit the vast majority. However you need to know what toilet you have as each almost universal part doesn't fit something another one will. Odds are you have a common toilet that any universal part fits - but there are toilets that are not compatible ith anything.

before doing anything though should you replace the whole thing is a question worth asking. do you prefer a round or engladed bowl; what about the height; maybe you want one without stains; or a low water use version that works? It wouldn't cost much more to replace the whole thing with one that fits you better.

Mostly, but as others have said variations exist. Another caveat is that newer/shorter tanks may need to removed to access the nut to change the filler valve.

@MajorHavoc has it covered. It's the flush valve that varies most. Most common is the flapper design, but some brands use plastic towers.

Fill valves are almost universally interchangable. Your variables there will be age and interference. You'd have to go back to basically antiques before things become incompatible. As for interference, units with the big float on the end of a rod may not fit in a tank that uses a tall flush valve. They make more compact styles for those situations.

All that bring said... What are you trying to fix? The flush and fill valves are by far the most common maintenance items, but if neither of those things are the issue then you're going to run into proprietary parts fast.

If the "kit" says it's "one size fits all" or similar wording, check the instructions that come with it on how to adjust it for things that are likely to vary.

Otherwise, you'll probably need to take measurements or, if you're lucky, there'll be make and model numbers on the existing parts that you can use to get replacements or new internal parts.

If you're really unlucky, you have a bizarre but genius-designed toilet made by a now out-of-business manufacturer who found novel and unique ways around patents and lawsuits and you'd be better off replacing the whole thing.

Not exactly universal but there is wide compatibility. If it looks the same then you are probably ok, but you need the right type.

Nothing new to add, but since crowd sourcing answers is more reliable when you have more of them, I figure it's worth it.

As everyone before this said, it isn't a perfect compatibility, so you can't just grab any random kit and be certain it'll be 100% right. But, there's a decent chance it will be, or that you can improvise things enough to get it to work long enough to get the exact right bits.

Biggest problem I've run into over the years is flappers not making a good seal, and the pipe not fitting well. The flapper is harder to deal with, but the pipe can usually be made to work with a gasket cut to size, long enough to get a better one at convenience rather than having to run right back out.

While the kits may use standardized plumbing connector, they are not all guaranteed to use the same standard sizes.

Not even national. I have a toilet with a ceramic flush valve built into the tank and it's almost impossible to find a flapper that's set up for this.