'He didn't get to swim' is an accepted translation, so does fick inte also connote that he didn't get a chance to swim or only that he was not permitted to? 'Did not get to ...' in English can be kinda ambiguous in that sense.
Question, in the tips and notes section in the table the classification is: får = may, be allowed to; få = get, receive, but isn't that the same verb only får is present form and få infinitive form? Maybe the definition "receive" should be under the får row in the table, this way it's a bit confusing (at least for me, don't know about others)
No, they are not the same verbs. They just happen to look and sound the same. The translations you refer to are correct. E.g. Jag får godis varje lördag. means that someone gives you the candy; Jag får äta godis varje lördag. means that you are allowed to eat your candy. As you notice the first sentence has one verb but the second has two, because "får" is an auxiliary verb.
How does one know which is meant? Was he not allowed to or did he just not get to (maybe he was running late)? Is the sentence unclear in this sense?
For me the correct translation showed up "He couldn't swim" which is totally wrong.
It's an accepted answer because in English, 'you can't' is sometimes used to mean 'you are not allowed to'.