A lot of the points raised as to why dogs aren't allowed in my opinion do seem a little bit contradicting in regards to the fact these don't apply on trains - i.e. - "Dogs aren't allowed on trams because of potential fouling" - well this is an issue on trains too. "Safety, damage to trams, delays, animal's behaviour and people with allergies" - while I understand these are possibly valid points (though I'd argue some of these aren't an issue or don't exist as issues) I don't see why they don't apply to trains too and therefore dogs shouldn't be allowed on those.
In my opinion I think the main factor is that the Metrolink is seen as a busy city light rail system mainly used by commuters and shoppers/leisure in and out of the city, on vehicles which are smaller than train carriages, and therefore they feel dogs aren't suited to this type of environment - in comparison to trains which often include much longer distances, bigger carriages, better staffed (?) perhaps.
There is therefore the point that, Manchester's more local heavy rail lines e.g local stopping services that are often packed at rush hour would also lie under this no dogs remit as these bare a similar environment to trams - but enforcing this on local stopping trains vs longer distance ones wouldn't make much sense.
I am a big dog lover myself, always had dogs, but I'd never bring one on public transport. One because I've never needed to, two because knowing me, something embarrassing would happen - if you can work out what that may be xD
In other words, I only really get the tram to head in and out of the city for leisure, shopping etc. therefore I don't really need my dog with me - but a TPE service to Scarborough for a weekend away on the train I'd be happy enough to bring my dog.