It's not a charity but a limited company
They aren't mutually exclusive. I work for a charity. It's also a limited company. It's pretty normal. Doesn't mean it's run for profit, does mean the liabiity on trustees is limited if it goes belly up.
Depending on how small this one is, it might not be able to register with the Charity Commission - you have to have an income over a certain level before they let you register.
Doesn't mean that it isn't dodgy, there's been lots and lots of small (and some larger) fraudulent charities over the years but for a small charity not to have a registered charity number doesn't make it so, and being limited liability definitely doesn't.
As for abbreviated accounts being filed, yes, there's exemptions to various reporting requirements for different sized charities (as there is for businesses) and there's nothing suspicious about one taking advantage of that. Bear in mind that most small charities don't employ an accountant, and it's likely that the accounts are prepared by one of the trustees in their own time, who may well not be a trained accountant either.
I have no knowledge of this particular one though, so I'm not offering an opinion on it's legitimacy either way. Quick look at its website gives no information on its trustees, its governance arrangements, links to annual reports detailing specifics on where the money's been spent so, for me, I'd probably want to know some more before I gave them any of my hard-earned, but a sloppy website might just tell you that they prioritise spending their dosh on their beneficiaries over "back office" costs, which is hardly a bad thing.