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Those drawers unless I'm mistaken are not on runners and are designed to be opened by pulling the top of the box.
Yes, the drawers run on a couple of pieces of what look like dowel. They’re probably not the best designed or constructed set of drawers, but they fit the shelving when we bought it. And it would be handy now to have more, they just not available to buy. They’re also tiny, I’m not sure runners would be useful as they’re so small.
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Muppetteer
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If you’re not worried about them looking the same, but just want some the same size, you have more options and a potentially easier task. It would be difficult to exactly match the veneer on the mdf anyway. Given the choice, I’d avoid doing a mitre joint. Assuming a dovetail is too much of a stretch, the front and back could be butted onto the sides and secured with dowels and a pair of cam and dowel concealed fixings, and glue. Ikea style. You’d need a jig for the dowels and concealed fixings. Still quite a bit of work involved.
Sharkstar
Looks to be 18mm veneered MDF or similar. Assuming it is 18mm a 5 X 3o domino would be fine as long as you double checked when setting the depth. Personally I'd use a biscuit as it would do the trick just as well and is cheaper, but the festool system is pretty much industry standard. Spline is a possibility across the mitre but would be mega fiddly as a concealed spline within the joint.
I doubt it end grain to end grain glue joints are, to say the least, weak. In engineered timber where the fibres are usually very short and the substrate is highly absorbent its a point of failure waiting to happen. Those drawers unless I'm mistaken are not on runners and are designed to be opened by pulling the top of the box. They'd be on the fixing pile within 6 months of purchase in my house; where all drawers are full to the brim of random heavy stuff and usually operated by heavy handed, overexcited children.