During the ceremony to gain an animal companion, is the animal summoned from a different plane, or is it required to be an animal that the druid has tamed ahead of time?
There's nothing in the Player's Handbook that talks about the origin of the animal companion, so I think we're in house-rule territory here. My recommendation is: the druid's class feature is that he has an animal companion, and we should try to interpret the rules in a way that doesn't make that weird.
So, for example, if the druid is in a very inhospitable location and needs to get a new animal companion, it's probably better to say "you summon your animal companion from a different plane" rather than "you can't get an animal companion because there aren't any animals here".
(If the DM feels that the druid character is too powerful, we can imagine the DM might deliberately say: "you can't get an animal companion right now because there aren't any nearby", just to get some respite from the animal companion dominating every battle. But interfering with the druid's class features like this is cruel, and should be avoided unless it's absolutely necessary.)
(This is a change from D&D 3.0, which explicitly said that you have to go find some animals and cast animal friendship to make them into your companions.)
If the druid has a specific animal which they want to make into their animal companion for roleplaying reasons, we may as well go along with that -- but please note, the druid's animal companion uses "the base statistics for a creature of the companion’s kind". So, even if the starting animal is unusual in some way, its statistics get overwritten with the base statistics when it becomes an animal companion.