-There are three aspects to it
For your question Galaxies have a production rate of stars per year (In the Milky Way 3 to 7 per year, but much greater in the past). Two planets can form from the same stelar cloud in two similar mass stars with the metalicity to produce terrestial planets in stable orbits within similar habitable zones, However this stars are likely to remain toguether for at most a few hundred million years. Now to have two solar systems with similar age and conditions unrelated to each other in close proximity is possible but very unprobable.
So my guess is that the chances of two planets near each other with similar civilizations is not very likely as per the fact of different starting points or resets to evolution due to extinctions. If the planets would be at a few thousand light years away it maybe more likely as per the stages in evolution of stars (you will find more stars of high metalicity). However this stars and planetary systems will be separated in their creation by tens to a billion years and again if habitability is reset in their evolution this makes predicting it not possible. However if enough space the whole galaxy is taken it is possible to find two planets that were created at the same time with similar conditions.
A-Biology reset by large cataclisms
Evolution even if with similar conditions on inception are the same it is 100% sure that both have had different and at different times large catastrophies so creating a difference in the timing and possibly also in the biologica development paths between the two worlds. Therefore biologically distancing them in terms of evolutionary complexity and possibly even afecting the types of biological entities.
B-Evolution of Brain size
The development of large brains as per number of neurons. (brains developed in three or four different stages and the history of it is burried in the structure of our brain. (A nervous system evolving to a primitive reptilian brain and then evolving into a mamalian brain). This last part happen arround 300million years ago. For this to happen all previous dominant species had to be extinct possibly because in this way large populations could develope and by means of adaptation and intermixing a new genera was created better adapted. Mamals have the longest life spands of animals on earth and also the longest expands in terms of maternity and the long periods of defensiveness of the offspring until maturity is achieved. (It is this long maternity - learning from parents, that created the need for large size brains so their offspring could learn from their parents - This is an smart way nature provided animals means of adaptation to the environement as adaptation possibly is better as it occurs across the life of an individual and is not as determined by the previous genes that were passed to you only although nature prudently keept our ancestral brains also in place so they interis very eact in creating adaptation.
Now if evolution has a set of rules (we do not know as we are the only example on earth of a path of evolution) However thre appear to be hints that there are laws to evolution. If this laws are the same every where that of adaptation it is very probable that life evolves in a similar way across the universe. Then if this is the case the containment of brain development is subject to population explossions, extinctions and the development of new species with traces of those prior with the hability to evolve rapidly large brains. In the Genus homo it took 46 million years of monkeys with arms to manipualte things to develop into apes and then 6 millon from this last to the present to grow brains our size. However across time there were 15 maybe more homo genus types with different brain sizes and each specie had a slightly larger brain from the few species only three or four developed suffciently large brains as to evolve into us. They all shared and imnproved certain technology along the way. Humans have 3x to 4x more neurons than the largest brains in the animal kingdom.
There is another caveat and is that genuis homo Primitive pre-civilization species tended to be small in numbers even after hundreds of thousands of years Only homo sapiens has multiplied exponentially in the past 50 thousand years and is the only surviving species. So we can see to that to develop a civilization a large population has to exists as to multiply the speed of complexity and thus speed evolution. So as long as there is no explosion in population of any single specie there will be no social development and technology even if large brain exists (case for Neanderthals). Even in the case of neantherthals they had their chance to develop agriculture in the previous interglacial period (130K to 120K years ago) but they did not. There is one issue with this and is social complexity as they were never more than 100,000 individuals at that time while in Homo sapiens there were 4 to 5 million at the time of the neolithic.
proximity as a relative term
A- Proximity in Space
However closeness is relative as per our understanding of the size of the unverse and above all the fastest method of transportation at one time. So 100 years ago we imagined selenites or martians, today habiatable worlds at 50 ly radious and in 10000 years maybe 1/4 of the size of our galaxy. So I would treat proximity with attemtion as it is dependant on our perception of our world.
B-Proxmity in Time
I think that two similar civilizations could only be able to recognize each other if they both have pursued similar purpose broad goals across their development which are derived from their needs. Could we say that those that were able to make fire a million years ago could recgnize a light bulb (sure) but they will find very difficult to understand how the ligh is generated. So this two civilizations will not be separated by evolution that mach. I think in one million years light may be still in use and possibly in 100 million years. So this is a partial answer as there may be some parts that could be undertood but others not. Could we say that this two civilizations are close in proximity yes because their needs and goals are the same (food, light, heat, raw materials, communication, medicines, etc). However if you expect to find two civilizations developing the plane that will brake the sound barrier at the same time you will not find them. (Space establishes relations between objects in space and time relates them in evolution).
However if we would find an extremely advanced civilization it is possible that all his needs will not exist in a more primitive civilization (as would be the case of meeting a computer machine civilization or a non phisical entity) in that case it is very difficult that they understand each other at all. This two civilizations will be separated by billions of years. So as long as there are biological entities they will recognize each other and their goals and needs aklthough it will become wirder and wirder.
A ultimate case would be that of an extremely advanced biological civilization that may reach a plateau lasting for millions to hundreds of millions of years where a civilization that dominates and perfects a technology has diminished returns from his development and has evolutionary limits on how it can achive this. Imagine for example a biological civilization that has been able to dominate and manipulate all the existing laws of physics. This civilization may find any further advance is halted by limits by the laws of nature or by the required engineering to an scale which is unachievable or that requires extremely long periods of time or unatainable resources In this case two temporary different advanced civilizations will converge at one point and their differences will became smaller and smaller as the more advanced will find it harder to progress while the younger civilization will get closer to the development point of that more advanced so they become similar. (However for an outside observer there will allways be a difference between the two stages of development but they will resemble more the closest the get).
C-Proximity in brain processing capacity
What is the procesing capacity of a civilization individuals - The difference of our brain with that of an octopus is 4x more neurons. (And the brain is already prety packed with neurons to the point of saturation). Now if as Paul Davis says biological evolution is only a brief space of time towards any biological entity building machines with incredible intelligence and being surpased by them as per intelligence and capabilities. So far machines have been better than humans in specifc tasks and it is understandable as our brain is the product of an extremely long time spand of well over a hundred billion sucessfull evolutionary steeps and trillions upon trillions of experiments on variable dynamic environements that yields the best possible evolution for survival.
So yes intermediate Kardashev_scale from I to III given the interpretation of similarity in time and that of space provided it will be able to recognize each other although we may find each other culturaly in oposite extremes.
Yes possible for the most advanced possible civilizations (hibrid or post biological or non even physical) as they reach a plateau of knowledge where further knowledge and technology progress is limited or capeed.