Custom aspect ratio LR Classic

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There's another option: you can use the Print module. This lets you define a page size as measured in inches wide and high. Within that page you can size an image cell (that your photo goes inside), also as measured in inches wide and high. So that defines the desired shape as well as the size of your final printed photo.

The way your photo goes into an image cell involves some options. First, the image can be auto-rotated to best suit, or its orientation can be kept.

Also, the image cell can be set "Zoom to Fit", or "Zoom to Fill".

FIT puts your current Crop inside the image cell as big as it can go just touching the sides. So the image will fall short and leave gaps whenever the shape of the current Crop is not the same as the shape of this cell. This is the same behaviour as when Exporting.

FILL achieves the same result as further cropping the image to exactly match the shape of this cell - only, it does so just on the fly and just for purposes of this particular output. The Develop crop itself does not need to change, just because you want to print the image one way or another way.

So with Zoom to Fill, the same crop can remain but be output however needed, to multiple shapes and sizes. Those shapes and sizes can be saved as print layout presets.

Each time you'll lose more or less image content at top and bottom / at left and right depending on whether each printing scenario happens to be a squarer, or less square, shape than your crop is. But the same would have happened, if you had instead updated the crop in Develop.

Print module can either output directly to a printer, or else to a JPG file which can then be sent to a printing company. The JPG shows the entire page layout at a desired pixels-per-inch resolution and colourspace selection - the required details for this should be sought from the printing company.

The print layout can be made borderless if desired, but sometimes this does not produce a true-to-scale result - depending on certain technicalities in the printing.

GoldingD is correct: Consistent with the nondestructive nature of Lightroom Classic, image pixels are not locked to specific physical dimensions until a copy is exported or printed, so cropping is just the aspect ratio, with the physical dimensions set at render time (print or export). This means you do have to treat the aspect ratio and the final physical dimensions separately. For example, you set the aspect ratio to 12 × 20, and then if you want a 12 × 20 inch print you would specify inches in the Print module or Export dialog box.

Deferring the physical dimension to when media-specific copies are made does make sense when multiple copies of the same image is reproduced for different media. For example, if after printing you then want to upload the same image to a web site to promote your print, naturally the web page image would be specified in pixels and not inches, so for that you would take the same original image aspect ratio but specify (for example) 800 pixels wide in the Export dialog box.

It does seem a little odd at first, but for me it’s become natural, because it isn’t uncommon to reproduce the same image at multiple print sizes. So it makes sense for me to maintain a single original image that is 4000 pixels wide, cropped to a specific aspect ratio, and only resolve those 4000 pixels to a specific physical dimension at the time a print is made (at 10 inches wide, 20 inches wide, etc.)

Although it isn’t possible to save a single preset that sets both the aspect ratio and final physical dimensions, the Crop tool does remember the last five custom aspect ratios used, and it is possible to make presets for printing (in the Print module, Template Browser panel) and for exporting (in the Preset panel of the Export dialog box).

Cropping in LrC (in develop) is not dimensional, it is a ratio. So 12x20 is not 12"x20" or cm, or pixel, or anything

As you can aspect, crop ratio 12x20 is the same as 6x10, or 3x5, LrC does not care, it is all the same ratio.

Incidentally, if you want to export to specific dimensions, make sure your crop matches your desired dimensions by ratio. So if you want 12"x20" in the export file size, make sure you crop to 12x20 (or 3x5, etc) otherwise odd things happen.