I'm not sure whether this is a growth chart, height chart or height measurement. Its photo is at this website.
The image depicts a "height ruler." To be more precise, it is a wall-mounted height ruler. Friendly decoration on a height ruler is common in a pediatric (kids) doctor's office. The photo shows a person taking the height measurement of a child.
A "height chart" is group of data points (height) taken over time and plotted on a chart or graph. Many families will make a chart of their children's growth by marking a door frame with a pencil and making note of the individual child's initials and the date.
Of course there is no reason that a person cannot use the depicted ruler as a chart as well by recording the height on the ruler. There is no way to tell if this is happening in the photo, but I don't see a writing implement in the man's hand.
Height chart would probably be the most accurate to my mind though I'd imagine some would call it a growth chart as it would be used to the child's height which changes as they grow up. Thus, I could imagine in informal conversation someone calling it a growth chart.
Height measurements would be more if there were marks on the wall at different points in time rather than just the scale. The photo is showing heights and thus could be used for measuring the height of a child.
Ruler would be a better term now that I see someone else's answer to this.
As many other answers have indicated, there are a couple of words you can use:
Despite this choice - natives would normally pick "height chart" or "growth chart" with "height chart" consistently more popular than "growth chart".
Sadly consistency is not prevalent in the UK or elsewhere - indeed some shops choose to sell both. Tesco (a major retailer in the UK) sells (at the time of writing) 7 height charts and 1 growth chart.
Here are the statistics for wall-mounted "X" on Google:
Height chart: 228 000
Growth chart: 122 000
Height ruler: 17 800
I think they're commonly called "growth charts". When I was a kid my parents had something like this on a wall and they'd write our names and dates as we grew.
The medical term for a height ruler that has an indicator-stop that rests on top of the patient's head is a "stadiometer." But it's a term that most native speakers of English never learn...it's jargon.