Brick can be quite expensive, with a price totalling more than buying a Birdies raised bed more often than not. The soil underneath the brick often needs to be worked on before you can even build on it to prevent sinkage and the likes, and not everyone has the experience nor knowledge to ensure it'll last for years to come, and not crumble after a short while.
On top of that mortar can contain nasty chemicals/contaminants which can leech into your soil over time, contaminating your plants.
All in all, it's much easier and more viable for most to invest in a steel raised bed instead, or to opt for wood.
If you have the time to spare (and a lot of clay soil on your property), one could very easily build a clay-brick one with a much lower cost than commercial bricks. Waste bricks from a demo might also be cheaper. The main thing will be ensuring the brick walls dont bow out once the fill goes in.
Unless the raised bed is quite low, like two bricks tall, and holding plants that don't have particularly strong roots, it would probably need to be treated like you're building a retaining wall to hold in the weight of the soil, which is a bit different to just building a brick wall and requires a bit more engineering knowledge to avoid the walls collapsing in the future.
Soil against a wall of any kind can be incredibly heavy. I'm currently in the process of having to build a small retaining wall along my side fence (which also has me digging out tree stumps this week) because some not-so-clever person in the past history of my house thought the fence itself would make a sufficient retaining wall and it's now falling over because that doesn't work even though the height of the soil pushing against it is only around a foot high (my yard ground level is higher than my neighbours) as we're on the side of a hill. I also have a 16 inch thick old stone wall up the back under threat of collapse due to soil being up against it up to 2 feet tall where my land rises up the hill, it's 100+ year old natural stone design is allowing it to literally bend under the pressure, but one day it's going to give and all fall down at once. There's another old slightly thicker stone wall in the same area that was completely demolished by 3 feet high soil pushing against it in which a now giant and overgrown jade bush was planted. So yeah, never underestimate the weight and pushing power of dirt and plant roots.
For this reason, building any kind of brick and mortar or concrete wall to hold back soil needs a bit of extra care and can only be safely done to a certain height (depending on type of bricks and mortar and design used) and requires a bit of engineering understanding of how much pressure any given type of wall can hold vs the pressure soil and plants can exert. Some places also have legal restrictions on how tall such a thing can be for safety reasons. Beds made from steel and/or wood, are a bit less dangerous if they do collapse because they do it differently. Mortared bricks and concrete rarely just crumble away slowly, they usually take a lot of pressure then one day go BANG and kind of explode as they collapse rather suddenly all together, a bit like a damn failure only with soil instead of water. Whereas steel and wood collapsing under pressure is like a disaster happening in slow motion... each day you can see them folding and failing and falling apart and dirt leaking out a bit more than the day before until eventually everything is laying on the ground and half buried under the soil.
There are people that have successfully made brick and mortar and concrete raised garden beds, there's even pavers and huge concrete planters designed specifically for it, so it can certainly be done if you know how. But as Mandy said, it can quickly get expensive. As JP said, if you have a lot of clay on your property that you can dig up, and/or access to second hand bricks, you can make your own for next to nothing. But that can be a lot of time consuming physical labouring work that not everyone has time for least of all the physical ability or know-how, but if you have it all, go for it.
Edit: All that being said, a narrow enough garden bed of brick and mortar or concrete is likely to be okay, because being narrow it's not holding a lot of soil. A wider garden bed would be better off with a thicker lower wall. My advice on the matter is look up how to build retaining walls and why they need to be built the way they are, then look at commercial garden beds and how and why they're built the way they are. You'll notice things like Birdie's beds once past a given size have structural supports running through them. Many wooden/wood-steel ones have structural supports that are buried quite deep into the ground. Mark not so long ago made a video about how he fixed up one of his home-made beds of wood and steel that was on a slope and starting to collapse.