At my home I have got a few different plants (Aloe, Rose, Neem, Pisum, Air plant, Lucky Bamboo). I got some type of soil (mixed with compost) from the plant nursery, and I'm wondering if using the same type of soil for all of them will be fine, or if I need to arrange for their specific soil requirement.
General purpose planting mediums are intended for a wide range of plants, and most will be just fine within it. The exception would be acid loving or lime hating plants, such as Camellia, which will thrive best in an acidic compost, usually sold as ericaceous potting compost. Other plants, like orchids, do best in a specialist orchid mix; large, permanent shrubs do better in a loam based mix, so the thing to do is check the requirements of plants you want to grow to see if they require anything in particular regarding growing medium. Succulents such as the aloe you mention would prefer a freer draining compost, but as already said, will do fine in general purpose provided they are not over watered or waterlogged.
I agree with Bamboo. I think a cursory search on google.com with the name of the plant and "soil requirements" will turn up the best solution.
While a large majority of plants you'll buy at a non-specialty store or nursery are just fine in a common potting soil, there are several that aren't and a few reasons why. One big reason, right off the bat, is that many of the potting soils like "Miracle Grow" are water retaining. This is supposed to be a benefit for people who don't water as often as they should.
For a plant like a "Peace Lily" or a "Petunia", which like to be wet, this a boon. For plants like a "Scheflera" or a "Dracaena", which likes to dry out between waterings, it's not as good. For succulents and cacti, it's down right lethal. It depends on the environment they're used to. It's also partially about oxygen. Excess water doesn't kill plants, because they can be grown in hydroponics systems. It's the lack of oxygen that the water causes that rots them.
That is the reason that you can buy specialty soils. These are typically just amended versions of the potting soil you'd buy for a regular house plant. Cactus/Succulent soil is simply amended with perlite, vermiculite, sand, etc... to make it drain more efficiently. Those plants like to be watered heavily, then dry out quickly and for a good period of time.
On the opposite end of that scale would be how you can plant carnivorous plants in sphagnum peat moss and leave them sitting in a tray of water. They do great this way and it's about the only way to use this media as it becomes hard packed when it dries out. They pack cacti in it at the box stores, so that they don't have to water them often. Try buying one and see how quickly it dies. It's because the peat moss becomes dry and hard packed. Water simply flows off the top, around the sides and out the bottom. If you soak it, the water holds against the roots and rots them.
Besides water issues, there are feeding issues. Plants require different levels of nutrients and what most plants might enjoy, will kill and root burn others. There are also plants that won't thrive if you don't supply additives. "Blueberries" and "Azaleas" for instance, like acidic soil and it's easy to add this to the soil for them.
Overall, it does matter what media you use in a broad sense. You don't have to sit there measuring pH, but you do need to get a soil that's in the right category for your plant and then look to see if it has special requirement. If you don't do this you could end up frustrated and give up on plants that you could enjoy for a long time.