Schottky diode not behaving as expected
You seem to be a bit confused as to what diodes really are, and what they do.
All diodes, including Zener diodes, act in a way that is analogous to one-way valves.  They only permit current to flow in one direction, but will block it in the other.  In the comments you mention something about Schottky diodes working &quot;between rail voltages&quot;.  That is not how I would characterize any diode, and I think you may think diodes are something they're not.
Diodes will conduct from their anode (the pin without the stripe/that the arrow is pointing away from) to their cathode (the pin with the stripe and that the arrow is pointing at) but only if the anode is more positive than the cathode.  If you connect one in series with a resistor, then connect the resistor and diode series pair one way round to a battery, it will either conduct about as much current as you'd expect with the resistor alone, or none at all depending on if it is connected the right way around or not.  Flip the series pair so the lead that was connected to + is connected to -, and vice versa, and it will do the opposite - start conducting if before it did not, or now draw now current when before it did.
All diodes have two important properties: Forward voltage, or \$ V_{F} \$, and the reverse breakdown voltage or \$ V_{BR} \$.
The forward voltage is the voltage differential that needs to appear from the anode to cathode for that diode to begin to conduct.  I like to think of them as having spring-loaded one-way doors.  It takes a little bit of effort to push open the door and hold it open because there is a spring trying to pull it shut.  That spring is the forward voltage.  This also determines how much power a diode will dissipate, as that voltage drop obeys ohm's law like anything else.  A diode with a forward voltage of 0.6V at 1 amp will be generating 600mW (\$ 0.6V*1A\$ ) of heat.
The reverse breakdown voltage is the voltage applied in from cathode to anode (so, in the opposite direction.  This is the direction the diode normally blocks any current from flowing) that will overpower the poor diode's voltage blocking ability and just force current through the diode anyway.
In the case of your Schottky diode, 5V is just the maximum reverse voltage it can block.  Exceeding this will cause it to begin conducting current in the direction it would normally block, and this current will be limited only by any resistance in series with the diode.  For many applications diodes tend to be used in, there will be little to no series resistance, so exceeding this breakdown voltage can often destroy the diode due to the high power dissipation it will typically experience.
All of this applies just as much to Zener diodes.  The only difference between a Zener diode and other diodes is that Zener diodes are doped such that they block reverse currents fairly poorly, allowing a little bit to leak through (called reverse leakage current).  They allow electrons to tunnel through them in reverse with much greater ease than regular diodes, which also results in their important property:  their reverse break down occurs much more sharply, and thus at a specific voltage.  This is what lets them act as voltage shunt regulators when used with a resistor.
You can, in theory, achieve this with other diodes as well, and as long as you limit the reverse current with series resistance, its fine.
However, Schottky diodes are a special case.  They have a very thermally sensitive junction inside which can be easily damaged by far less reverse current flow.  Even with the series resistor, it is possible that your Schottky diode is now damaged.  It also might be ok, but just so you know.  Schottky diodes should never be used in situations where they might undergo voltage breakdown (or, put simply, how you were using it).
Beyond that, other diodes, while tolerant to being used this way (unlike Schottky diodes), are not well suited to this usage case.  Non-zener diodes do not have nearly as sharp of a voltage breakdown.  This property is also generally specified only as a minimum and not controlled beyond that.  The specific breakdown voltage can only be expected to block up to the rated value, but the actual voltage it will breakdown at will vary by volts or more from diode to diode.  The breakdown itself will also happen much less sharply, so increasing the current through it will also cause a noticeable increase in the voltage across it.  A Zener diode, on the other hand, thanks to the much sharper breakdown, maintains almost a constant voltage even over a wide range of currents, allowing it to act as a shunt regulator.
Besides Zener diodes, there are regular or rectifier diodes, which are general purpose and just regular old diodes.  Schottky diodes work via a completely different mechanism as rectifier and Zener diodes, and as a result, have much lower forward voltages.  This makes them more efficient than rectifier diodes, especially for higher currents.  They also do not have something known as reverse recovery.  If a zener or rectifier diode is conducting in the forward direction and suddenly experiences a reverse voltage bias across it (like, for example, AC current reversing direction - or trying to), it doesn't actually block the reverse flow immediately - it takes a little bit of time to turn off, and will allow current to flow in the opposite direction during this time.  The time is typically very short, but it can result in substantial efficiency losses, heating, and other problems in certain applications.  Schottky diodes do not have this problem - they do need to turn off because they don't turn on either.  They act as true one-way streets for electrons via thermionic emission.  You will still see data sheets specify things like reverse recovery for Schottky diodes, but this is entirely due to the blocking action acting like the two plates of a capacitor, and as such a Schottky diode only allows a fixed capacitance worth of charge to conduct in the opposite direction.  This can be an important benefit, depending.
There are other, more exotic diodes with more specific uses, but rectifier, Schottky, and Zener are the common ones.  Well, and LEDs of course.  LEDs behave, for all intents and purposes, like Zener diodes.  Ones that light up when current is flowing forwards through them.  As long as you control the current with series resistance, you can even use LEDs in place of a Zener diode if you're in a pinch.  Some circuits, like a certain 3 component tesla coil circuit, famously use an LED like a Zener diode in this way.