Supports usually want utility which directly helps their mates
There are quite a few times when supports get items either for stats or for utility that gives the team some extra functionality. For example, Morellonomicon is a very popular AP support item, and a lot of tank supports get Randuin's. Neither provide amazing utility, but the personal stats are very desirable. That's not really something that is applicable to the case of the portal, but I do want to bring it up.
As for the portal, think about it like this: Say I'm a tank support, or even a utility support. Say that the enemy has very strong waveclear and we don't have good tower sieging mechanisms. Heck, just say that my team is putting a lot of effort into teamfighting as opposed to slpit pushing. I am likely to have boots of mobility and some downtime when other people are farming. It costs me little to move to another lane, lay a portal that threatens their tower, and get back to my team quickly enough to prepare for a teamfight. I can do this easily while warding up. This can either get us a tower, or force enemies to go deal with the problem, at which point we have an advantageous fight. It's a strong active, and supports do buy items with mediocre or inefficient stats in order to get strong actives (see Talisman, Crucible, Twin Shadows instead of Morello's, etc. This also frees me up to get more stat-efficient tank or AP items because my CDR is covered, and that is very desirable in some games.
For this portal, this question will have to be asked: Why should I have the tank buy an item that gives them inferior stats and leave them to split push, when I could just have the support lay the portal down instead? A stat-based increase in the portal's power gives a clear answer to that, because the portal is stronger when someone with gold and stat-based itemization places it. For CDR, I'm not sure that there is a good answer.
Like I said before, I don't think this is an all-or-nothing kind of thing. I think CDR could easily be part of the item, and I don't think that it's a matter of it either being good on supports or it being good on tanks, but I do think that having it give inefficient stats with no reward for certain types of itemization does encourage it to be built on supports more often than on junglers or laners.