Does data rate increase with channel frequency?
Is it a misconception that higher frequency channels support a higher bit rate or is it true for some reason?

It is not the (carrier) frequency itself that determines the supported bit rates but the available Bandwidth of the channel

Suppose I have 10 MHz of bandwith available at 100 MHz, for example 100 MHz to 110 MHz

or 

I have 10 MHz of bandwith available at 1000 MHz, for example 1000 MHz to 1010 MHz

Then the highest bit rate I can achieve will be the same as there is 1 MHz available in both cases.

However note how at 100 MHz that channel bandwith is 10% of the carrier frequency but at 1000 MHz it is only 1 %.

At that 1000 MHz I could fit 10 of those 10 MHz channels to come to the same 10% (or use a wider channel of 100 MHz.)

If we want divide a certain frequency band between service providers, that's much easier to do at the higher frequencies. It can be done also at lower frequencies but that would result in narrow channels (small bandwidth) and therefore lower bit rates.

To put it more simply: there's more "space" at higher frequencies so it "costs less space" to implement higher bit rates.

Suppose I go to 10 GHz, that will give me another factor 10 more space.

The 2.5 GHz Wifi vs 5 GHz Wifi isn't completely fair because the 2.5 GHz band is about 100 MHz wide (that's all channels together) but the 5 GHz band has about 900 MHz available (depending on your country it might not be one continuous 900 MHz range though). See here. So there's simply lots more space (bandwidth) assigned to the 5 GHz standard.