My mom has this plant in her coastal Southern California yard. She cuts it back to just about nothing every year and it grows back super healthy looking.
Here's a picture of its leaves and flower.
That's definitely a Plectranthus, specifically looks like Plectranthus scutellarioides (formerly called Coleus) - they're super resilent and can be cut back hard each year, which is why your mom's plant thrives with that treatment.