Someone told me that this effect doesn't stack and only one perk will work. Is this true ? I have been searching on google but I can't seem to find anything about it.

I don't have a definitive answer but this should be fairly straightforward to test.

Set up two perks, one which applies a fire spell, the other applies an ice spell, ideally use spells that have a duration.

Give perks to the player, test in game, and use the console to see whether that target has effects with keywords MagicDamageFire and MagicDamageFrost

Yeah I tested it and it unfortunately doesn't seem to work.