Why can't maximum zoom (ie the change in size of an object on screen) consistently be stated on compact cameras? I am aware of the complications with focal length and lens size but I am only interested in how much the camera can magnify.
The most useful specification for comparing how much of the image frame you can fill with a distant subject is a normalised maximum focal length. For historical reasons we consider an image sensor the size of a 35mm film frame to be "normal", so this specification is quoted as the "35mm equivalent" focal length.
For example, a Sony RX100 VII is advertised as having an 8x zoom, with a 24-200mm "35mm equivalent" focal length range.
By comparison, the competing Panasonic Lumix LX100 II is advertised as having a 3x zoom with a 24-75mm "35mm equivalent" focal length range.
These focal length equivalents are (largely) comparable, so the Sony will enlarge a distant subject to fill (200/75) = 2.67x more of the width (or height) of the frame. You can compare the specifications of other cameras in the same way.
Consider a superzoom camera like a Canon Powershot SX70HS. With an advertised 65x zoom and a 21-1365mm "35mm equivalent" focal length range, it will fill (1365/200) = 6.8x as much of the frame with a distant subject as the Sony RX100 VII, and over 18x as much as the Panasonic LX100 II.
However, this extraordinary range does not come without compromises. This telephoto reach is achieved by pairing a lens of modest focal length with a tiny image sensor.
All things being equal a larger image sensor gives better image quality than a smaller one, and the sensor in the Panasonic is around 7.5x as large as the one in the Canon superzoom giving it a dramatic advantage in image quality.
Similarly, it is worth mentioning that in general lenses with larger zoom ratios are optically inferior to those with modest zoom ratios. In professional camera lenses, ratios of around 3x are common because in this range a lens can be designed for very high quality while retaining a reasonable size and weight. "Prosumer" tier lenses may have zoom ratios up to about 11x while retaining a size/weight/quality tradeoff acceptable in that market.
Superzoom cameras, with their attention-grabbing ratios spanning ranges of 30x, 65x, even 125x, generally have very poor optical performance in addition to their small sensors, and ultimately create a highly compromised image.
Likely what you are after is to cause the software of the camera display magnification as you zoom. Further, you would like this value to be equal with the power as related to binocular or telescope. However, you need to know that most authorities deem such a value valueless. This is because our modern cameras are miniatures meaning the image of objects will likely be a reduction (not a magnified view).
That being said, we can categorize the zoom, wide-angle, “normal”, or telephoto. So let me try to explain. The industry considers the “normal” view to happen when the focal length setting is approximately equal to the corner-to-corner (diagonal) measure of the format. Try looing up your camera’s specifications to find out the size of its imaging chip. If its is a compact digital, it will be 16mm height by 24mm length. The diagonal of this rectangle is about 30mm. Thus in this case, a zoom setting of 30mm delivers a “normal” angle-of-view. The angle of view will be about 45°, camera held horizontal (landscape). This is considered “normal” or magnification 1 as compared to binoculars. This is not without caveats. This is because we will view the camera’s miniature image on a computer monitor or a print on paper and these procedures yield an enlargement. In other words, the additional magnification complicates. Nevertheless, a focal length about equal to the diagonal measure is usually considered “normal”. Not to be confused by macro work where magnification 1 is a life size image projected by the lens onto the digital image sensor. To achieve, the object is placed about 2 times the focal length away from the front of the camera.
OK we’ve got “normal” down, now for wide-angle, this will be a setting of about 70% of “normal” or shorter. For the compact digital, this will be 30 X 0.7 = about 20mm or shorter. The realm of telephoto is about 2X normal. For the compact digital (APS-C format) that 60mm or longer.
To put this all together, you need to know what the “normal” focal length for your camera. Gleaning this data might take some research on your part. If your camera came with a Kit zoom lens. It will have a range of likely 18 thru 50mm or the like, you can bet that the center of its span will be OK to consider as “normal”. Next imprint fact. Each time the focal length doubles, the image size projected by the lens doubles. Conversely if you zoom out 2X focal length, the magnification drops by ½. Suppose you are using an APS-C with a 240mm lens. What is the equal magnification compared to binoculars? “Normal” of this format is 30mm. Zoom to 240mm and the magnification is 240 ÷ 30 = 8 (about equal to 8X binocular). Nobody said this stuff is easy!