Can I write that I had a phobia in my personal statement?
It is a judgement call. On the one hand, if you are sure it is no longer relevant, then mentioning it could be a problem since some would doubt you are cured, I think.
On the other hand, it shows personal growth. Congratulations.
But having grown as a person may not be as relevant to future research as you think. At least in the minds of others.
On balance, it seems best to not discuss it in written materials. If it comes up in an interview, it is again a judgement call whether it is wise to mention it. But, at that point, you will have more information on which to make the judgement.
A personal statement is a story/pitch that describes/sells your suitability for a position. Your historic phobia of animals might be a starting point for your story. Overcoming that phobia showcases part of your personality.
Do you think its a good idea to include that I overcame my phobia by myself?
I don't think overcoming your phobia is the key message, I think how that fits into the broader story is. Try writing a story that starts from your phobia and concludes with your application. See if that story can be worked into a personal statement you're proud of.
Buffy mentions the possibility that hiring someone with a historic phobia of animals might be a red flag for a position with animal studies. I agree (and up voted), but I think it depends on the broader narrative.