The problem with that is that animals have to have an intelligence of 1 or 2:

Intelligence score of 1 or 2 (no creature with an Intelligence score of 3 or higher can be an animal).

To speak, you need an intelligence of 3:

If you have a penalty, you can still read and speak your racial languages unless your Intelligence is lower than 3.

So if you do that, the companion PC is no longer an animal. It's a magical beast, and thus not eligible to be an animal companion.

You could house rule away that restriction of course, but I don't think it's a good idea. If the PC in question wants to play as a magical animal that happens to be the Druid's friend, that would be an easier way to go about it. That lets him play a fully realized character with his own will & ability to interact with the party, instead of what is functionally a trained animal. You could use the monster advancement rules to let him gain levels, or let him take class levels if he wanted to become a Cleric Unicorn or something.

Not by RAW

Simply because the rules written, which you have quoted, say nothing about it being such. It doesn’t mention Leadership or Leadership’s cohort at all, and while it mentions similarities between Wild Cohort and Animal Companion, it does not actually state that Wild Cohort is or counts as Animal Companion.

It’s more than reasonable to houserule it, though

Allowing someone to qualify as having an Animal Companion or Cohort, because they have the Wild Cohort feat, is perfectly reasonable. I have done so in many games, and it’s never been an issue.

In this case, though, I would not actually state any interaction between these things. If you are a druid or ranger and take Wild Cohort, you just get both an Animal Companion and a Wild Cohort, as separate creatures. Same, too, if you take Wild Cohort as well as Leadership: you get a regular cohort as well as the wild one. The two don’t interact with one another.