Does high inflation in the UK and US have the same cause?

There is a tendency to explain inflation in terms of the policies of a particular government but given the similarity in the rise in the UK and US, do they actually have the same cause?

Not all causes are the same but broadly speaking they are very similar. Inflation is positive change in the price level and the price level is determined by equilibrium in money market.

There are various ways how money market can be described, but a simple, and for this case sufficient, way of doing so would be just using standard textbook New Keynesian model. In such model money market is given by:

$$\frac{M}{P} = L(Y,i)$$

where $$M$$ is money supply, $$P$$ price level $$L$$ money demand that depends on output $$Y$$ and interest rates $$i$$.

Taking logs of both sides and taking time derivative we can show that:

$$\pi = \frac{\dot{M}}{M} - \frac{\dot{L_Y}}{L}\frac{dY}{dt}-\frac{\dot{L_i}}{L}\frac{di}{dt}$$

where $$\pi$$ is inflation (positive change in price level $$\frac{\dot{P}}{P}$$, $$\frac{\dot{M}}{M}$$ is change in money supply over time and $$\frac{\dot{L_Y}}{L}\frac{dY}{dt}-\frac{\dot{L_i}}{L}\frac{di}{dt}$$ is change in money demand over time which depends on changes in real output and interest rates over time.

So using the standard NK model above we can see that inflation in a country depends on a tug of war between money supply and money demand. The model is bit oversimplified, in real life expectations have non-trivial effects and so on but for answering your question this model would broadly suffice.

So lets have a look whether the variables affecting money supply and money demand were evolving similarly in UK and US.

First, when it comes to money supply we can see from Fred data that money supply as measured by $$M2$$ increased in both countries (data for US; data for UK). So when it comes to money supply side of the tug of war the root causes are the same, even though naturally the money supply did not expanded in both countries by equal measure.

Second, let us now examine the money demand side. When it comes to real output we can see there was decline in real GDP in the US in the last quarter. In the UK in the UK GDP has still increasing trajectory but the data is also half a year behind the US data (see Fred data for US; data for UK). Also, both economies just barely recovered from Covid19 recession (US more so than UK) and are expected to be hit hard by the Russo-Ukranian war so it is fair to say that in both countries real output is expected to decline significantly.

Now here when it comes to causes of fall of real output some of the causes will differ. UK output was likely negatively affected by Brexit, whereas US might have been bit more affected by clogged ports and supply chain issues.

When it comes to nominal interest rates we can see that they were both kept low until the end of 2021 both in UK and US (see data for US here; data for UK here). There was recent hike in this rate (since likely both central banks are concerned about high inflation and want to combat it) but the hikes were small, and inflation statistics are reported with lag so they are not necessarily reflected in the numbers you cite.

To sum it up, the causes of inflation in US and UK are broadly similar. Especially when it comes to money supply and interest rates. When it comes to real output one could find more differences in the original causes between UK and US, but they are still having effect in the same direction.