Yes, they would.
I can affirm this because mounts follow the general yet extremely potent principle of specialization : A specialized entity -person, machine or animal- will always perform better than a generalist or worse, an incompetent one at the task it is specialized in.
The very concept of giving and sharing tasks with others -and by extension domestication- relies on the idea that you cannot do every tasks as well as at least another one : Biology constraints, experience and knowledge, all of these make you specialized for something, and weaker in others... Than others. After all, you cannot be good at everything! Combining merfolk aptitude at planning and resource management with carefully chosen pet skillsets will create synergies and give generally better outcomes.
To focus on mounts, here are a few hard advantages that mounts often have over their riders. Hard because, there's very little which can match those without excruciating costs behind or some modern machines. So now grab your whale, dolphin, seahorse (ok, perhaps not them) and go for this ride :) :
Maximum power output and sheer mass
Even in water, heavy house stones and tightly packed kelp balls will still be very heavy to move around. Like on land, take a big animal for these kind of tasks.
Also, have you seen an armoured horse charging at you? Worse, a war elephant? A metal-plated cachalot could break through any enemy lines with ease. Heavy mounts are great for frontline wars.
A mount can be faster than you. Unless your merfolks are moving very fast, they'll ride a mount. If you have big black marlins, I wouldn't be surprised to see merfolks attached to them running races, too1.
When we basically do anything from walking to pushing stuff, we use our own energy. You can move this energy cost to a third party for two main benefits :
First, you can sit back and relax, enjoying a nice cup of Yorkshire tea while your mount does the job, Attach a carriage to a giant medusa, throw it in the Gulf stream and you're in for a gentle, peaceful ride along the flows.
They act as energy batteries. Use a pair of walruses to travel from Kelpington to Coral City, then exchange them for another pair to reach New Beach's town. You travelled twice if not thrice the distance you would have made alone, with very little efforts from your part.
Because not everything is physical, animals are also known to help people get through hard times or to live better with their mental disorders. Dolphins could be a good alternative to horses in that regard : smart enough and very social, sea?
Also, having a mount displays the social status of someone : If you manage to feed your 5 orcas, it means you have a good chunk of meat and money in your palms. Any people of high-status would have one mount just to show they're comfortable financially. In other words, rare or tough to tame mounts can make great positional goods.
1 : Well... I would be bubbling out loud of surprise by the whole scene, but not because of their speed :p.