I wouldn't worry about it - Space Coast Aviation is owned by Pan Am Flight Training which is actually now owned by All Nippon Airways the Japanese airline.
A lot of the schools operate like this and so do a lot of non aviation related companies for various business reasons. Even CTC have subcontracts for part of their training and I would hazard around 50% of schools have some sort of slight name difference.
Your responsibility : Know who your paying money to - and pay by credit card so you can reclaim it if necessary.
It's all a mute point if they can't sponsor you for a visa.
If we are strict about how we read the EASA licensing rules, if in their opinion the training is provided by Pan Am and not Space Coast, that training does not count towards your license. The rules are too young and they haven't really been tested, but I would exercise caution every time there is even the smallest ambiguity about the ATO/training provider credentials.
Its not a moot point, there are visa types which allow training that are not sponsored by the school. Being approved by TSA to train non-residents and being SEVIS approved to issue I-20 are two completely separate items.