was being kept illegally. was kept illegally.
They both seem to fit in the past progressive tense to me.
If you have a link to a good explanation, that would be appreciated.
No, the tenses are actually different. The tense that's being used in the first sentence fragment is called the past continuous tense (formerly known as the past progressive tense, as far as I know) and the tense that's being used in the second sentence fragment is called the past simple tense (also widely known as the simple past). Both of your examples are in the passive voice (Modern English has two grammatical voices: the active voice and the passive voice). For any further clarification, just follow the links that I provided.
There is nothing progressive about "was kept". There is something (at least close to) progressive about "was being kept".
was kept -- past tense, passive voice, indefinite aspect
was keeping -- past tense, active voice, continuous aspect
was being kept -- past tense, passive voice, continuous aspect
When a state remains the same for some time and no progress is made, "progressive" seems a strange label to use -- but it is often used in exactly that way. I prefer to use the label "continuous" for expressing a state with duration, regardless of whether anything changes.
All three of these phrases are in the past tense. That tense is marked by "was".
In both passive voice cases, some form of the verb to be is followed by a past participle form. This is the pattern that both your original examples follow. Those two combinations are "was kept" and "being kept".
In both continuous aspect cases, some form of the verb to be is followed by a present participle form. Only one of your original examples fits this pattern. I added "was keeping" to compare to the "was being" example that you provided.
The last example follows both the passive-voice pattern and the continuous-aspect pattern. It contains both "being kept" and "was being". Yes, that one word "being" is a part of two overlapping patterns -- once as a form of to be and once as a present participle form. It is, after all, the present participle form of to be.
If we want, we can have more than one overlapping pattern in the same verb construction:
had been being kept
form of to be + past participle = passive voice
had been being kept -- past, perfect, continuous, passive