The big complication here is that an individual soldier might have have a lot of equipment allocated to them which isn't necessarily in their possession or immediately available to them at any one time.
One example is NBC protection equipment. For example a soldier may be issued with a respirator and a protective suit, respirator filters, decontamination kit, medication etc etc but there may be any number of additional sets of the consumable items held in storage either at unit level or more centrally.
Equally the equipment an individual soldier needs will vary according to their role and the climate they are operating in. This is well illustrated by the British Army's 'black bag' issued to troops deployed in Afghanistan which has three different protection categories plus specialist equipment for armoured vehicle crews and this is just personal items and clothing and doesn't even consider weapon systems or other operational equipment.
In general training the equipment allocation may well be comparable to what an air-softer would own as practicing basic infantry skills doesn't really need that much expensive kit. Indeed there was a period when a British infantry soldier might well have used very nearly 100% privately purchased kit on exercise (apart for weapons and helmet), indeed in the 80's and 90's optimizing kit was practically a hobby.
The overall, qualitiy of issued equipment ahs also markedly increased, as an example these boots (£50) are the previous generation wherrease these are current issue (£134) in fact now soldiers even get a choice of brands.
There is also the problem of how you account for things like food and ammunition. While you could well consider general catering to be a running cost there is an argument that at least some quantity of field rations could legitimately be part of their operational equipment. Ammunition is even more difficult. As a baseline there is a certain amount required even in peacetime for training and marksmanship practice and on active operations an infantry soldier could expend very large and very variable quantities. At the very least a rifleman could be expected to be carrying 100 rounds and possibly a lot more. 6 30 round magazines plus belts for a section machine gun is certainly not particularly excessive and in some cases additional boxes or loose rounds or bandoliers of stripper clips. Many armies also use section or platoon level 2" mortars or increasingly 40mm grenade launchers.
The best example of individual weapon costs I can find is the L129A1 ordered as an urgent operational requirement to improve the range and accuracy of firepower available at section level, at a cost of 1.5M for 400 units.
Equally in the last decade of so the amount of technology used by ordinary infantry soldiers has increased dramatically. Most individual weapons now have optical sights (which need to be a lot more rugged than civilian equivalents) and it is increasingly common for every infantry soldier to have a personal radio. Also officers and senior NCOs may well be issued with laptops for base administration, longer range radios as well as GPS and other tactical electronic and communication equipment. The use of body armour and blast protection has also dramatically increaced in the last decade or so.
This BBC article suggests that the cost of the current generation of body armour is around £1400 per unit.
It is also not necessarily a valid assumption that military equipment is cheaper to procure than civilian equivalents. For a start military equipment often has complex standards and requirements which need to be met and in many cases they won't be buying equipment 'off the shelf' but will often end up covering the full development costs of a particular item. For example the SA80 is reputed to have cost around £450M to equip around 150 000 troops and it also had a fairly major upgrade a few years ago at a cost of around £500 per weapon. This also means that in cases like the AR15 system the government may have effectively subsidised the civilian version as a civil version has already had most of its development and tooling costs covered by military procurement budgets.
Equally even quite small infantry units now have access to some very expensive weapon systems, for example one-shot anti-armour/material missiles can be getting on for £10,000
So a lot really depends on exactly how you account for spending and it's not really that meaningful to come up with a 'per soldier' cost in any case as any individual unit isn't much use without the logistics and support arms which allow it to operate. For example the cost of an infantry platoon calling in one airstrike could very well dwarf the cost of the equipment they are carrying with them. Equally not every single infantry soldier will be issued with every piece of equipment they would need for operations at any one time so many of the costs are per item not per soldier.
Overall it is probably safe to say that the actual personal equipment and uniform that a soldier is issued as their personal 'property' may be only a few hundred £/$ but once you start to try to account for operational equipment held at unit level it really depends on your definitions, for example do you include organic assets like vehicles and crew-served weapons held at company/battalion level ? Do you include the costs associated with storing and shipping equipment to an operational theater ?