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I can do that (U1A, U1B...) if all blocks are in the same schematic sheet.
In Tools > Annotation > Annotate Schematics, make sure "Complete Existing Packages" is set to "Whole Project".
In general, though, I would argue that you should keep the SOM footprint/symbol separate from the footprint(s)/symbol(s) for the connector(s). There are several reasons for this, but the big one is that Altium can't produce multiple BOM entries or placements for a single part. So you would need to add separate BOM/placement entries somehow for the SOM and however many connectors it needs.
Instead, consider making a footprint for the SOM itself that has an overall outline with reference geometry for where the connectors should be placed on the PCB (preferably references you can snap the connectors to). Once the SOM footprint and matching connectors are placed, optionally union them in the PCB to make them a little easier to handle. This means your BOM and placement entries for the SOM and connector(s) will be generated automatically, and makes it easier to manage the connectors and their footprints separately from the SOM footprint*.
This doesn't (easily) allow you to use multi part schematic symbols for logical connectivity blocks--but you can place the SOM connectors on one sheet, and then use harnesses to bring out the various sets of HDMI/I2C/USB/whatever signals out of that sheet to wherever you need them. This is significantly more flexible anyway, since you don't need to overhaul the SOM schematic symbol to use different sets of pin functions. If you want to show what signals are on what positions on the SOM, you can include that information in the schematic symbol. In fact, you could create the SOM symbol with pin labels (but no pins) that you can drop your connector symbols right next to, such that the connector pins line up with the signal names.
* Consider that if you want to switch from, say, PTH headers to SMT, you can do that by just replacing the headers, with no need to edit the SOM. Similar if you need to edit the footprints for any reason: you can just edit the connector footprints, and don't ALSO have to edit any SOM footprints where those connectors are embedded.