Which kind of passive tags can be found up to a 25 meter distance?
Not too many. To be able to extract data from a tag (ignoring passively powering it) you would need the tag to "reply" using a proper radio technique such as UHF. To power a passive tag you can use either a radio waves (such as UHF) or by NFC (using magnetic coupling).
If you want to know about powering a passive tag (at distances of 25 metres or so) using UHF techniques, then you would need a directional transmit antenna to concentrate the radio wave power towards the tag. If you try and use an omni-directional transmit antenna (like a dipole or monopole) then you will transmit a lot of unused power into space. Think about how much light power a lamp produces and what miniscule fraction of that light energy is received by the eye.
But, if you are interested in exploring powering NFC tags (at distance) then, this is what my answer concentrates on. It's all about gleaning as much magnetic energy as possible to produce sufficient DC voltage to run a tag for a few tens of milliseconds.
What kind of reader is necessary?
Regards the tag responding to the reader, that's more problematic; at distances greater than a metre or so, it can probably only be accomplished reliably using regular UHF transmission.
Maybe also read this Q and A on UHF RFID Read Ranges - it covers the types of RF power levels and antenna needed to energize a tag at distance.
In the main...
This answer is about how you can acquire sufficient energy to power a tag using the magnetic near-field. If tag and reader are close, then remodulation of the near-field to respond to the reader is viable. But, firstly you need to power the tag.
What the blog article doesn't say is important
What is not disclosed in the article are any details about the transmit and receive coils. For instance, in the plane of a 10 cm radius coil carrying 1 amp, the B field (magnetic flux density) at the centre of a single-turn loop will be 6.28 μT (micro teslas) and this is certainly a level that can be turned into electrical energy that can power a passive tag device quite easily. This is easily established as is shown further down.
How much flux density is needed?
At 15.5 cm distant from the above loop, the B field will drop to around 1 μT and we're probably on the edge of being able to convert that field density into usable volts to power a passive tag with a small sized receive coil so, take that as a benchmark when reading the rest of this answer based on my experiences.
A bigger transmit coil and more current
If the radius of the reader's transmit coil became 1 metre and, the loop current was increased to 10 amps (easily done using capacitive resonance without significant power losses) then, the distance from the reader's transmit coil to achieve 1 μT flux density is now 1.55 metres.
if this technology really exists with a 25 meters range, and if it's really available for home DIY projects, or if it's still a
professional high-end technology that requires high-price investment and/or weeks of engineering to make it work
As a comparison, a 10 metre radius transmit coil with 100 amps flowing will produce a flux density of 1 μT at a distance of 15.5 metres. Now 100 amps sounds a lot (and it is) but, with high grade capacitors and a decent driving circuit it can be done (but it ain't cheap).
A little bit about the calculations I'm doing
The following picture is an extract from a web-based calculator on HyperPhysics: -

It's definitely worth having a play around with it.
Receive coil size and number of turns
Up until now, I've regarded the passive tag as having a "small sized" receive coil but, there's nothing in the blog that suggests this is the case (other than the assumptions we might make). What I'm getting at here is that when we calculate B (magnetic flux density) it's just a number we can use for making a comparison. It's the total flux through the tag's receive coil that is important when liberating enough energy to power the device.
Receive coil circuitry
So, if a small sized receive coil was assumed to be 10 cm radius, we could easily convert to "area" and make a not-unreasonable assumption that B was fairly constant over the whole coil area and, calculate the voltage induced. Then, with a decent tuning capacitor that voltage could be magnified by about 30 times (experience allows me say this), rectified and used to charge a capacitor to power the tag for a short time (a few tens of milliseconds).

Above picture from this related Q and A
But, if the tag's receive coil was 1 metre radius, we would have a capture area that is 100 times bigger hence, the induced voltage would also be 100 times bigger. If the receive coil also had ten times more turns, the received voltage would be 1,000 times bigger. So now, we can "adjust" our expectations of a viable B field amplitude and assume that it can be 1,000 times lower at 1 nT. This is now the "new" flux density benchmark in this answer.
So, a 10 metre radius transmit coil powered with 100 amps produces a flux density of 1 nT at a whopping distance of 184 metres!
Sanity check on transmit coil current
But, we know we don't like to run 100 amps through the transmit coil so, if we dropped that back to 10 amps (perfectly feasible without Litz wire or hyper expensive capacitors), 1 nT is present at 85 metres distant (still a very reasonable distance): -

We might want to reduce the transmit loop size a bit to maybe 5 metres radius and the distance that we get 1 nT flux density is now 53.7 metres. With a 2 metres radius we can get 1 nT at a distance of 29 metres.
Does 1 nT and a single-turn 1 metre radius receive coil cut the mustard?
- The coil area is \$\pi r^2 =\$ 3.14 square metres.
- 1 nT at 29 metres from the transmitter will be constant across the coil aperture.
- Hence total flux at receive coil is 3.14 nano webers
- If we operated at (say) 1 MHz, \$\dfrac{d\Phi}{dt} = 2\pi\times
10^6 \times 3.14 \times 10^{-9}\$
- This is 19.7 mV per turn hence, ten receive coil turns produces about 0.2 volts RMS.
- This is a decent level for extracting energy to power a tag.
Conclusion
- So, is a 2 metre radius, single-turn transmit coil carrying 10 amps all that unreasonable - not in my book. It's a perfectly normal transmit coil in many applications.
- Is a 1 metre radius, ten-turn tag coil all that unreasonable in some applications - I don't believe so. It's a less common scenario but there will be some applications.
- Does the blog article say anything useful about coil sizes and currents - not really.

It's used for tracking parts in factories over a large area: really, even with passive RFID tags? Can you maybe elaborate about this in an answer, it's exactly what I'm looking for. – Basj Nov 24 '19 at 21:49