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I'm not a cat person. I've never been a cat person. I can feel no affection for a haughty prima donna equipped with claws and fangs. So I sympathise with Helena, the protagonist of this hilarious one-woman show – whose life is invaded by a hellish creature determined to shred her carpets and drive away her friends. Yet that's just the starting-point for an unexpectedly touching story, which explores big questions of connectivity and identity under the cover of a lightweight comic tale.

There is, of course, no real cat on the stage. But at times I had to remind myself of that, because actor Elise Heaven makes her invisible companion utterly believable – wrestling with a crate as it tries to escape, pulling back and forth in a tug-of-war over a pasta bowl. And it's not just the cat; other men and women have walk-on parts, each evoked every bit as strongly. It's a rare treat to see a solo show so fully-realised, or for a cast of imagined characters to so completely fill the room.

There's plenty of physical humour in Heaven's performance: my favourite moments included an undercover mission with a hoover, and one of the funniest drunken nights out I've witnessed in years. But her self-penned script delivers its share of entertainment as well, with unexpected one-liners and a series of comic scenarios building a compelling picture of domestic warfare. There's a gentle parable at work here, about caring for something or someone you just can't understand – and predictably enough, as time goes on, we all develop a sneaking affection for the diabolical feline.

In the same way as that relationship develops, a story that first seems fun but inconsequential morphs into something far more. We know from the beginning that Helena was born in Slovakia, a fact that we – like she – regard as no big deal. But as the calendar rolls forward to the Brexit shock of 2016, that once-innocuous detail becomes a key factor in her future life. And suddenly, the jokey story about giving house-room to a cat is revealed as a powerful metaphor: for what it means to have a home, what it's like to belong, and how cruelly that certainty can be denied.

This is a play about fitting in, in all sorts of ways; about somehow finding the round hole to slot your own round peg inside. The ending is beautiful – if a touch improbably neat – and the insight on life it shares is ultimately an uplifting one. Stylish, hilarious and impeccably performed, Passpawt still finds time to deliver an impressive philosophical punch. It might not have turned me into a cat person… but I left the theatre feeling subtly changed.