The mod Portable Mobs adds a Basic Capture Cell and a Master Capture Cell. Right-clicking or shift-right-clicking with one of them on a mob grabs the mob and "puts it" in the cell, e.g. updates the cell's NBT to store it. Right-clicking with the cell places the mob again. There's also probably other mods with pokeball-style capture items as well.
https://www.curseforge.com/minecraft/mc-mods/portable-mobs

For mobs with a right-click, such as villagers, shift-right-click usually works as a workaround. However, the human companions have a secondary behavior with shift-right-click, the stay/follow toggle, and it takes precedence over the capture. If it's not too much trouble, could that behavior be disabled when the item in the player's hand for shift-right-click is portablemobs:* so that a capture could work? (basic_capture_cell and master_capture_cell)

Thanks, and I would understand if it was too much trouble.