What is the evidence to claim that political order in ancient Rome was sufficiently different under "kingdom", "republic" and "empire"?

Interesting question but ultimately the answer is no. There are also various continuities between Muscovite Russia, Peter's Empire, the Soviet Union, Yeltsinite Russia and Putin's Russia but they are really not all the same thing.

You are comparing a small city state and a big empire in the timespan of about one thousand years. The difference between what was Rome in the beginning and what was eventually the empire is too big.

When Rome was a kingdom it is obvious that the kings could not be too powerful. The small population and the productivity of the primitive agriculture of that age could not support a big bureaucratic machine and a big army. The king was a little bit more than village chief. Directly in contact with the population.

Thus it is quite difficult to see how the Roman kingdom was not somewhat like a modern presidential republic

Small Rome back then was quite different from a modern republic where a complex system of check and balances is required and often is not enough to keep in check the big bureaucratic machine.

The event known as expelling the kings was in fact redistribution of powers between the magistrates.

Of course. Back then there wasn't the level of organisation we reached today.

Under republic some new offices raised that were not less powerfull than that of rex. Those are of military tribunes and dictators. Arguably Roman dictator was even more powerful than a rex.

The story of the Republic was not uneventful, they went through period of revolts and periods that we might call of martial law. Due to the long wars, but also the internal instability.

Under republic (and possible under kingdom as well) the title "Imperator" was quite frequently used to refer to military leaders.

When Rome never became formally an empire, it reused the titles used during the republic and even during the empire the emperors formally where the chiefs of the army. Nonetheless the concentration of power was in practical manner was very different.

Under dominate order it was theorized that there should be four emperors at a time, a practice unusual for a monarchy.

In China they used to say the emperor is far away and the mountains are high. The size of the empire with the slow communications of the age rendered the control very difficult. So, it was just an expedient to shorten the command chain, but it didn't work very well since co-emperors often fought for supremacy.

The senate always retained theoretical power for electing new emperors and disposing the ruling ones.

Most of the times when the emperor and the Senate confronted each other the legions sided with the emperor. The number of times that the senate had to accept and vote in an emperor proclaimed by the army is much greater. Again, there is a big difference between the formal structure and who holds de facto the power and also this changed frequently in and empire (the Western) that lasted about four centuries.

So what are the reasons to ascribe certain periods in Roman history to monarchy and certain to republic?

The practical situation on the ground.

What is the evidence ... ?

The sequence of events showing where the power was. E.g. Few lines above I pointed out the most of the times the emperor was proclaimed by a legion and the senate vote was just a formality.

But mind you, don't assume that the roles were starkly defined. Often senators acted as counsellors of the emperor and some of them used the position to become quite powerful.

I believe that the main difference in these epoches is not the formal powers of involved offices or titles, but which social layers have what influence in politics.

In line with that I would say that the creation of the tribunates was a result of the conflict between plebejans and patricians. Due to the expansion of the Empire between 300 and 0, the raise of slavery etc., these social classes ceased to be the ones that defined Roman society.

The major change from the Republic to the Empire was the decision to keep for life as the head of state first Caesar then Augustus. Thus it put the top job in the hands of one man as long as that man was alive. There was no peaceful way to remove the Emperor at this point and since they controlled the army (or they would not be emperor) they had military power over the rest of society.

The change from Kingdom to Republic was maybe more subtle but put two men in charge instead of one. Thus allowing for a balance of power at the top. Again, this is a break with the previous regime.

In a similar way, you could say that the 4th of 5th French Republics are the "same": president, prime ministers, ministers, parliament, elections, yadda. Or that the French and English prime minister have the same function. Overall, you would not be too wrong but the devil is in the details. Thus why we refer to the Roman Empire, the Holy Roman Empire and the Byzantium Empire as different beasts even though they are all called "empires" and come from the same root.