For instance, I know that in the Roman Empire, many people would speak both Latin and Greek, along with a more more localized tongue such as Coptic, Syrian, Thracian, Gaulish, etc.
Also, to clarify the definitions I'm talking about:
Pidgins: A very basic mixture of simplified languages, which usually develops in places where there is trade between two parties that do not speak the same language.
Creoles: Typically these are Pidgins that have evolved into a full language, to the point where they are spoken by a community in day-to-day life.
Mixed languages: A semi-controversial category that may or may not exist, depending on who you ask. This is a language that uses elements from two separate parent languages, in roughly equal amounts - to the point where it can be considered as descending from two different families.
And I have one final question: if there are not any examples of these types of languages in the distant past, then why not? Are these categories of languages somehow unique to the modern world?
In effect all languages are mixed. There is no such thing as a pure form of a language, as all languages undergo divergence and convergence when in proximity to others, and borrowing/exchange often happens as well.
We should also remove pidgins from the list. Put simply pidgins are not languages but rather first-generation making-do forms of communication. They do not have fully developed syntactical rules. They can become creoles after a generation, which (again put simply) happens when a new generation grows up speaking it and from that things like proper syntax develop. There are a number of languages which are called Pidgin (Tok Pisin = Talk Pidgin), but these are not pidgins anymore. They are creoles. There are also a number of languages called Creole (or Kriol etc). These are fine for the purpose of the question.
Also, Pidgin itself is a pidgin word, coming from the pronunciation of "business". Pidgin English was business English. In many places, there were regulations to discourage or prevent either the occupied from learning the occupiers language, or vice versa. At times in China's history, it was a punishable offence to teach foreigners Chinese. Thus things like pidgins come about, where (for example) the Cantonese labourers in Shanghai have learned the phrases needed for basic communication with the occupiers, and vice versa. But it was strictly for business/trade.
As for mixed languages, better to just lump those in with creoles as well, if not just a matter of code switching.
Anyway, on to the main question: Yes, they certainly did. But generally not as creoles.
Like I said at the start, all languages are mixed. When a group conquers another, traces of the conquered language often find their way into that of the conquerers. This is called a stratum. After the Norman Conquest, English had a Norman superstratum. When Sinitic speakers moved into areas like the Yangtze delta, their language picked up a She substratum (among others). When various Romance languages influence each other, such as the spreading of /ʁ/ (that French R sound), this is called an adstratum, and adstrata are often the context in which areal features develop. These are features which spread over a large area across potentially unrelated languages and is a form of convergence. Tone in languages in Southeast/East Asia is an areal feature; languages which had no tone system developed tone due to contact with other languages which did have it, not because they took the tones (these are not always related languages with similar words to begin with), but just under the influence.
If you look at the way that modern Sinitic languages developed, it is through these sort of strata. Non-Mandarin languages in the Sinitic family have been Mandarinising over the years. If you look at the development of modern Romance languages, this is also what has happened. French sounds like French because of substrate languages. Middle English sounds like it did because of a substratum. These aren't creoles though. Instead they are forms of a language which was influenced by another, which resulted in changes.
To be a creole, first it would have to start as a pidgin, a make-do form of communication, which then developed into a language. That's not what happened with French or Hakka or English.
Pidgins did develop, and some remained, such as Kriol and Gullah, but this is not the norm in expansive empires with a central government in control of things. The reason for that is that, for the most part, major empires always push for language standardisation, and if there are the resources, education of the standard. Enforcement of standards means that you're more likely to see a case of the standard with a substratum of the occupied groups language, rather than a creole that is a mix of the two.
I hope that helps answer your question. Let me know if you have follow ups.
Full disclosure: I'm a historical dialectologist & documentarian linguist, not a creolist. There are people whose entire careers are looking at creoles. As mentioned, there are things like mixed languages which are controversial and which I don't really want to get into for that reason. If a creolist wants to come along and address that topic specifically, word.
Also, feel free to poke around r/linguistics where we do have talented and knowledgable creole specialists who can go much deeper into the details than this.