It is being done a lot all around you. With a lot of caveats, however. And not exactly doubling, but adding more like 50% of bandwidth.
So lets start from beginning:
First lets consider the ideal physics text book world, where cows are circles, pi equals to 3 or infinity (depending on who wrote that book), wind does not blow, humans live in vacuum, objects are completely rigid and antennas are ideal (meaning they are sensitive to exactly one ideal polarization).
In such a world if you have stationary (usually called fixed) transmitter and receiver antennas, you can make your antennas at transmitter and receiver aligned ideally. Then you idea makes a lot of sense!
What happens if one (or both) ends of the link are mobile? Well, in this still ideal world people will hold their mobiles phones perfectly vertical, of course!
Hopefully you start to understand the issue now...
So let's get back to this problems riddled real world now:
- Antennas are no longer ideal - meaning they are sensitive to all polarizations at once. Some are stronger, but still all of them add up. You can look up term "cross-polarization discrimination" (I'll call it XP later) about this.
- People are no longer holding mobile phones completely vertically. If they would be able to hold them completely horizontally, we could just have a switch inside a phone to swap polarizations, but these pesky people rotate their phones all the time in all the 3 axes! Consequence - your antennas will have a completely random crosstalk between polarizations!
- A bit better for fixed links... but pesky installers will never bother to align your antennas perfectly, wind, temperature changes, solar heating gradients on the structures will still cause trouble.
- As mentioned in another answer there are also reflections, but let's not dive that deep, that effect is usually small. So you need really long distances or very complex environments for that to have a significant impact. However, to be precise, here on Earth there is this pesky atmosphere instead of ideal vacuum, and various elements in it (ice crystals in clouds, hydrometeors (fancy word for snow, rain, etc)) will rotate your polarization! You must take this into account when designing satellite communication links!
So is it completely impossible? No. It all depends on the number called signal/noise (written as C/N, more specifically signal/(noise + interference), but that distinction is not relevant here) ratio.
This ratio defines how many times your wanted signal needs to be stronger (weaker) than noise for you to be able to receive it. It is normally defined in decibels (dB).
Let's assume you need 4dB C/N for you to receive your data. If your antenna has 10dB XP, it is much more capable of discriminating that polarization you want from the other and it makes sense to use both polarizations at once!
But real life happens again and in it C/N depends on your data rate and bandwidth. Higher the data rate - higher C/N is needed. Narrower the bandwidth - higher C/N is needed. There is a theoretical limit on how much data you can transfer per time unit within a certain bandwidth called Shannon limit (capacity, theorem, etc).
So of course you want tens if not hundreds of Mbps from your radio link! Then you'll need C/N of more like 40dB. And you want this in your mobile phone, of course! Then your antenna's XP will be more like 3dB. Oh, now we have a problem!
A bit side note - as frequency spectrum is very crowded these days, bandwidth is a scarce commodity. So for modern 4Gs, 5Gs and other Gs one can only try to get as close to that limit as possible within a bandwidth available. Needs to be said that we are extremely close to that theoretical limit, modern systems come like 99% close to it! And your phone actually uses multiple polarizations at once (still called MIMO, along other words)!
But there is a price in complexity of such systems. Your simple mobile phone uses most of tricks available for that - adaptive data rates, channel estimations and reconstructions, dynamic antenna pattern shaping from multiple antennas and others even more complex ones. Yes, your "simple" mobile phone may have from 2 to 8 or even more antennas in it (somewhat depending on phone's price, of course)!
By adding and subtracting signals from these antennas your phone tries to extract as much of single polarization signal's C/N as it can. It frequently exchanges data about signal quality with base station so they in tandem adapt data rate exchanged (thus adapting required C/N for successful data link). Base stations have even more tricks up their sleeve - they use antennas composed of hundreds or thousands of small antennas. Similar to your phone they use AI to phase shift signals to each single one of those antenna cells so that signal emitted from that station has actual beam formed towards your phone! And they change that beam form hundreds of times per second because they need to talk to multiple mobile phones within that second (maybe even couple of phones at exactly same time)!
Well, I think this is deep enough as an answer to your question :)