This fan token has an important role to strengthen the club community. Basically, fan tokens don't have to have a base value (or the crypto asset that they are based on) bitcoin or ethereum. In other words, the value of fan tokens is measured by the value of fans who buy tokens and the value of loyalty to the club they like. and when a fan has shown their loyalty to a club then the token will explode.
I see it like buying a jersey, or a ticket, or whatever merchandise you can think of, it is just buying something that belongs to your club and you are a fan of it so you want a piece of it, nothing more. Anyone who purchases with bigger ideas would be disappointed for sure.
There are some clubs who does extra stuff, cheaper season tickets, cheaper jerseys, chances to participate in some club events etc etc which could be great, you just need to hold certain amount and you would be able to do these, and if you are really a big fan, then you may want to get some of that token to do this, but aside from that I do not think that there is any other investing related logical reason to buy them.