My guess would be that fungal spores are fungal spores, and as such can affect anything in the tank - both fish and frogs alike. Just a guess, though. . . it could very well be that some are more prone to infecting fish than frogs - I'm not really sure!
I think your froggy should be okay, as long as he is healthy - with no wounds or abrasions, in clean water, and not under a lot of stress. . . In my understanding, various fungal spores are always to be found in any tank, it's only when the animal is under stress, as you say your betta is/was, that their immune systems falter and such things get the upper hand . . . I'd still do extra water changes to clear out any potential nasties - just to be as safe as possible - clean water can't cause any harm, for sure!
In the last year or so that I've been keeping them, I've lost only one of my frogs - my first. She had a few previous health concerns, and I made a few mistakes with her in the beginning. One day she got a really bad fungal infection, which I suspect was caused by past damage taking it's toll, though she had been fine for many months prior. I moved her into a QT/hospital tank, but in spite of all I did to treat her, she died. The other frog that shared the same tank was perfectly fine, and is now living happily in my 55. He had been kept well since I brought him home, and has always been a happy little boy. Not a fish/frog situation, but why I think that as long as he's healthy, your frog should be okay. . .
I wish I had something more concrete to offer you. . . hopefully one of the others will have come across a similar situation and can be of more help. I hope your betta recovers quickly. . . :(
Last edited by Chesh; 05-21-2013 at 05:12 PM.