For your case, one "line_ids" field should be readonly and suggest you to make readonly which line_ids is show fewer fields.

Because if you add data into Tab 2 line ids, it will also effect to Tab 1 line_ids.

Only Tab 2 can able to make add an item.

I make some changes below and please try again.

<notebook>
    <page string="Tab 1">
         <field name="line_ids" readonly="1">
             <tree>
                <field name="field1_model_abc_line"/>
                <field name="field2_model_abc_line"/>
                <field name="field3_model_abc_line" invisible="1"/>
             </tree>
        </field>
    </page>
    <page string="Tab 2">
        <field name="line_ids">
            <tree editable="bottom">
                <field name="field1_model_abc_line"/>
                <field name="field2_model_abc_line"/>
                <field name="field3_model_abc_line"/>
            </tree>
        </field>
    </page>
</notebook>