Scale of graph axes

must both axes use the same number of significant figures

Axes do not have values, let alone significant figures. If you are referring to the “ticks” indicating the scale of each axis, these do have values but they are exact, so the concept of significant figures does not apply.

Of course, if the graph has other labels that do contain experimental values, each value should be expressed with the appropriate precision.

As Brian's answer already mentions, precision is not a part of the coordinate system with its axes, labels, value ticks and spacings.

If I painted a portrait of you, it would be much less accurate than if an artist did. Even if we painted on the same canvas. It is not about the canvas. Precision and accuracy is only purely about the stuff you put on the canvas.

A coordinate system is your mathematical canvas. If you do a measurement and take into account the proper number of significant figures, then it means that you aren't sure of the exact value of digits and decimals beyonds this number of figures. You might also indicate such inaccuracy as a standard deviation or with +/- indicators or the like. They all symbolise that the actual value could be anywhere within this range.

Drawing the data value on a graph thus requires you to decide on how. You want to draw just a point, but you don't have just a point, you have a range. So what can we do? Possibly, you could draw the point in the centre of the range and then add error bars. Or you could draw a "shade" or something else creative to indicate the range "around" the centre. Possibly you skip the point altogether and only draw a shade. Possibly you skip the shade all together and only draw the point.

If you draw nothing but the point, then noone will ever know what the precision is - the axes don't tell. The canvas itself does not include any information about the picture you are painting.

When displaying experimental data on a graph, must both axes use the same number of significant figures

Each axis indicates a measure of some physical observable. Not every observable can be measured to the same degree of accuracy. Hence it's not neccessary to indicate the same number of significant figures on the two axes.

This question is touching on different aspects to the meaning of precision in graphing values. Let's consider only one axis.

Let's put experimental data onto a graph line. In today's world, we use a computer tool (e.g. a spreadsheet program). We have no inherent concern about how best to put the data point on the graph. A value of 10.0 and a value of 9.9 will be positioned automatically by the software application. The separation between the two points in the reference frame of the computer software will be controlled by the precision (NOT accuracy) of how the software stores a value. Software that only uses integer storage will be unable to distinguish between the two values. Software that uses floating point storage will distinguish the two values to the precision of its internal storage.

Let's now try to plot that graph. The separation of the two points here is the resolution of the device. Resolution is controlled by the precision (NOT accuracy) of the plotting device. All else being equal, an old style TV screen with only 640 pixels x 480 pixels will have greater difficulty positioning the two values for distinct resolution compared with a 5k resolution monitor device.

Finally, let's try to read the graph. Clearly, the resolution that we discern is controlled first by the resolution of the device where the graph is being displayed. Let's assume a perfect (analog) display device with effectively infinite resolution. In this case, the resolution is controlled by two things. It is controlled by how well we can see (discern) resolution, e.g. how well our eyes work. Let's also assume that our eyes are perfect.

This leaves us with a core answer to the question.

In a perfect world, the precision (NOT accuracy) that we can apply when we read back a value from a point on a displayed graph is controlled by the resolution that we can obtain using the scale markings on the graph itself.

For convenience and simplicity, let's assume that the scale markings on the graph have infinite precision (NOT accuracy) in their position and that we can read those scale markings with infinite precision (NOT accuracy).

The best precision that we can report in reading a value back from any graph with tick marks (scale markings) is half of the separation of the tick marks. PERIOD. Why do I stress this with PERIOD? Because in some readings, the implication is given that we could/might/may read to 1/10th of the tick marks. We cannot. A graph with tick marks at every one unit has a precision (NOT accuracy) of $pm 0.5$ in any value that is obtained from that graph using those tick marks. This is independent of how far separated the tick marks are located. We can only tell precisely that a person on a soccer field marked in 10 m increments has moved in $pm 5$ increments, no better.

Can we do better? Yes, often we can. In the digital case, we may do better if we can directly read back the digital value that is being stored instead of trying to read from the graph. For example, suppose that we have a digital image with a width of 1024 pixels and we are reading along an axis marked with 5 tick marks. Suppose that we have a value located exactly at 10/3 on this axis. Our report from the graph tick marks will be for a position at $3.5 pm 0.5$, a relative measurement (reading) precision of 14%. Our report using the digital pixels scale will be $683.0 pm 0.5$ pixels, a relative measurement precision of 0.07%. When we have no access to the (perfect) raw digital data, we can improve the given tick mark resolution by adding our own well-calibrated tick marks to the existing tick marks. Suppose by example that you printed the digital image out on a 10cm x 10cm page. The tick marks are separated by 2cm. The relative resolution will not improve if we use them to find the value. But take a millimeter ruler and lay it along the axis. Add tick marks for the millimeter positions. Using your new scale, you can report a position along the axis of $333.0 pm 0.5$ mm, a relative measurement precision of 0.15%.

Finally, why do I continually emphasize the phrase "NOT accuracy" in the above? Because the accuracy of a value on a graph is not established or to be addressed within the scope of your question. The accuracy of a value is established by making a comparison to the truth, not by how well we can read the value before we make that comparison. We report the precision of values that we read, not their accuracy.

In summary then, to increase the relative precision for reading a value back from a graph, rather than adding more digits to the numbers on the tick marks on the graph axis, add more tick marks along the graph axis. The presumption is that plotting devices today have more than the resolution you could ever want to support using as many tick marks as you might ever want.

I assume you are talking about the scales on graphs. A scale consists of a series of values associated with it, which placed near tick marks (ticks) on an axis.

The scale values will typically be round numbers, since round numbers are more readable, and since the ticks can be placed wherever is most convenient. (For example, you would not use values 13.6, 14.6, 15.6, you would use 13, 14, 15, 16). Adding extra zeros on the right hand side of the values (thus increasing the number of significant figures) is typically not done - the value placed at the tick is assumed to be exact. For example, you would not write 5.000, 6.000, you would simply write 5, 6.

How many significant digits are needed will depend entirely on the range of the axis relative to the value. For example, an axis running from 0 to 9 will need one significant figure to represent this. An axis running from 57.03 to 57.08 will require four significant figures.