Looks like what is known as an 'adjustable cup and cone' bottom bracket. That means the bracket can be disassembled, and the axle and ball bearings will come out; also the clearance between the cups and cones (the surfaces the ball bearings run on) can be adjusted. This differs from a 'cartridge' unit that is pressed together and cannot be disassembled or adjusted.
It's highly likely that the unit does not need to be replaced. It probably just needs adjusting to take the slop out of the bearings. You could disassemble the bearings and re-grease, but if the bike has not seen heavy use that may not be needed. What I would do is adjust the bearings then if the crank runs smoothly leave it alone. If the bearings feel rough, 'notchy' or if there is excessive friction, then look at servicing or replacement.
Park Tool has a good page an video that shows how these things come apart and how they are adjusted: https://www.parktool.com/blog/repair-help/bottom-bracket-service-adjustable-cup-and-cone
You can see that several special wrenches are required, and different tools are needed depending on what type of unit you have. In addition you will need a special crank-puller tool to get the cranks off the BB axle tapers. To put the cranks back on you really need a torque wrench to do the crank nuts or bolts up to the proper tightness. I have found my local bike stores will do bolts up with a torque wrench for free if they are not busy and I buy some small items.
If you are looking at replacement a cartridge unit is an option but of course you will need another set of special tools to install that. See https://www.parktool.com/blog/repair-help/bottom-bracket-removal-installation-threaded
Whether you replace the BB with a cartridge or adjustable unit one key measurement is the axle length, and whether the taper is JIS or ISO standard. This page describes the difference https://www.sheldonbrown.com/bbtaper.html. The bottom bracket shell is almost certainly standard ISO 1.370" x 24 threads/inch. The other key measurement is the bottom bracket shell width. The standard for road bikes is 68mm, mountain bikes 73mm. Sometimes the wider shell appears or road or hybrid bikes.