Can I fertilize aquarium plants in a similar way to garden plants?

In short - yes you can, in the market there exist a lot of various fertilizers with specific macro- or micronutrients. And yes, aquatic plants are not really different from terrestrial plants in terms of nutrient requirements.

However, both in case of aquatic and terrestrial plants, it is a little more complicated. For example, it is true what you stated about nitrogen - its deficiency causes insufficient chlorophyll production (condition also referred to as chlorosis) that makes the affected plant's leaves lose their green color; they get yellow or even pale white, starting from the tips. It also makes the whole plant look sick and thin - that's because severe nitrogen deficiency stunts growth and causes general starvation of the whole plant. If the plant's leaves are showing signs like this, then applying nitrogen fertilizer would improve their condition - but only if these signs were caused by the nitrogen deficiency in the first place. If the plant is already healthy and gets enough nitrogen, then adding nitrogen based fertilizer wouldn't make the leaves any more healthy nor green. It might cause plant to grow faster, though - but only up until a certain point, until there is enough nitrogen that it's no longer a limiting nutrient.

It also wouldn't help if the observed signs were to be caused by the deficiency of another nutrient; deficiencies of other nutrients - like iron - could also result in chlorosis. It might look similar, but could be distinguished by subtle differences. For example, chlorosis caused by mild nitrogen deficiency begins with the oldest leaves first because nitrogen is readily mobile inside the plant and thus plant tries to save itself by moving nitrogen from the oldest tissue to other places where it is more needed. On the other hand, chlorosis caused by iron deficiency affects the youngest leaves first because iron, in contrast to nitrogen, is immobile and cannot be transferred like that; also, the veins of the leaves in iron deficient plant stay green, while nitrogen deficiency causes leaves to uniformly lose their green pigment. An article about nutrient deficiencies in aquarium plants in general could be found at aquariumcoop.com.

What I mean through the last two paragraphs is that the growth of a plant is controlled not by the total amount of available nutrients, but by the amount of the scarcest nutrient available (it is a specific case of Liebig's law of the minimum). In this context, I think it is wise not to overdo with fertilization based only on a single nutrient for aquarium plants; also, some nutrients are antagonistic, like magnesium and calcium - the excess of one of them could cause deficiency of the other. More information about nutrient interactions in plants could be found on cannagardening.com (it is about garden plants, but this mechanism applies to all plants). If you are unsure what nutrients might be missing, there exist all-in-one formulas that incorporate three main macronutrients (nitrogen, phosphorus and potassium) together with trace elements.

Please also note that fertilizers support all plants, including algae, thus it is important to use them in a way that doesn't create new problems while solving the old ones.

Fertilizers like this are in a form of concentrated liquid that you add to water. An exact usage formula is written on the label, telling to add a certain volume of the liquid based on your tank volume to reach desired concentration. They could also exist in a form of solid pellets intended to be stuck in the substrate. Nitrogen containing fertilizers exist, but using them might not be really needed because decomposing waste usually generates more than enough nitrogen - both to feed all plants and still manage to raise in concentration over time (in the form of nitrates) to be pushing you for periodic water changes to get rid of it. I could see that using them could be useful in fishless and heavily planted tanks that utilize more nitrates than they generate, but otherwise it should be enough just to decrease the frequency of partial water changes to increase nitrate levels and thus avoid nitrogen deficiency in plants. I don't think it's common to have a problem with too low nitrate levels in aquarium, but it's definitely possible.

Phosphorus (in the form of phosphates) is also released into the water column from decomposing organic matter, but I am not familiar with whether it is enough for plants or not in freshwater aquariums. One source that I found about this - an article at aquariumcoop.com - tells that aquariums with high bio-load have tons of available phosphorus and nitrogen. It may be good idea to check phosphate levels with specialized test kits to tell whether you need to apply phosphate fertilizer for your plants.

In case of potassium, the third NPK plant macronutrient, the same mentioned article says that it is often the only limiting macronutrient of high bio-load tanks, and thus worth supplementing by applying a specialized fertilizer. Depending on the source, the optimal potassium level in freshwater aquarium is said to be from 10 to 20 ppm.

And also, as aquasabi.com notes, it is important to take CO₂ in account in case of aquatic plants:

In the case of any deficiency in aquatic plants, you should first consider carbon dioxide intake before dealing with the other nutrient factors. Most of the damage patterns listed below, such as yellow leaves or crippled growth, can also be caused by a carbon deficiency. CO₂ fertilization is essential for healthy plant growth.

Lighting could also be the cause of insufficient plant growth if the light spectrum being emitted is missing in regard to certain wavelength intervals. Plants are green because they absorb and derive energy from red and blue parts of visible light spectrum. If the lighting being used is of low quality, it could have "holes" in the emitted spectrum and not emit the most efficiently absorbed wavelengths; this could be the case in cheap LEDs and fluorescent bulbs because, in contrast to incandescent bulbs, their spectrum isn't continuous. High quality LEDs are good to use, though.

An article like at jbl.de about optimal concentrations of various nutrients in aquarium water could be used as a further reference for the general topic of fertilizing aquarium plants.