The linked method of "Real Induction" is not actually induction (though as a logician, I might be being a pedant about the use of the word "induction" here...). Instead, it is an application of the topology of the reals.

Loosely, what the "Principle of Real Induction" is saying is "if I have the bottom of $[a,b]$, and I can always get bigger, then I know I have $[a,b)$. But if I also have closure, I must have $[a,b]$." This is a slick way of proving properties about intervals, if we know we can always make our set of "good" points grow upwards. It is *not* induction, however, because we are adding LOTS of points at a time. When we make the move from $[a,x]$ to $[a,y]$, we are adding in all the uncountably many points inside $(x,y]$.

"True" Induction (the logician in me wants to remove the scare quotes, but the decent human wants to leave them in) is characterized by doing things one step at a time. Think about the natural numbers: We prove it for 0. That lets us prove it for 1. That lets us prove it for 2. That lets us prove it for 3. and so on.

The way to actually induct on the reals (actually the real polynomials, but it's the same idea) might be as follows. This isn't necessarily a good example, it's just the first I came up with.

Say we have $\mathbb{R}$, and we want to make it so that every polynomial factors into linear pieces (and pretend we don't know what $\mathbb{C}$ is...). One overkill way of doing this might be to take the set of all polynomials $\mathbb{R}[x]$ and well order it (using the axiom of choice). This gives us a good notion of the "first polynomial", followed by the "second polynomial", and so on. Then we'll go one at a time, and add in all of the roots of each polynomial. At the end of time, we'll be left with a field in which every polynomial has a root. I'm glossing over a fair number of technical details, most notably we have "limit stages" in the induction,
but these are not much of a bother.

This is not to say that the "Real Induction" you've linked to isn't useful! Quite the opposite. In fact, here is an excellent PDF with applications of this (badly named, imo) technique. It can clean up a lot of analysis in a very pleasing way. Indeed, it seems like many mathematicians enjoy using this tool, and think it is aptly named.

More philosophically, it is a fact of mathematics that different people will have different opinions about what is or isn't an intuitive name for an object or technique. Often, the names we choose align with how we view the relationships between different branches of math, and this can shed some light on how we think. I would never call this tool "induction", because it doesn't rely on a well ordering for its use. However I can see the similarities, and how somebody with a more analytical background might view it as a very similar tool in spirit, which justifies the name.
This is part of the beauty (and, occasionally, pain) of working with mathematics.

Let me know if you have any follow-up questions. I'll be happy to explain more, or clarify anything that you feel I haven't addressed.

I hope this helps ^_^