Please check the following code snippet, everything looks fine:

new Vue({
  el: "#demo",
  data () {
      return {
        selected_title_color1:'',
        titleColor1:'',
        colors:['紅色','藍色','黃色','粉色','土地色','水藍色','灰色','淺綠色','橙色'],
        colors_dic:{紅色:'red',藍色:'blue',黃色:'yellow',粉色:'pink',土地色:'blawn',水藍色:'aqua',灰色:'gray',淺綠色:'palegreen',橙色:'orange'},
        modalVisible1: true,
      }
  },
  watch:{
    selected_title_color1: function () {
      this.titleColor1 = this.colors_dic[this.selected_title_color1];
    }
  },
})

sssccc


      

        
{{ titleColor1 }} - {{ selected_title_color1 }}
      


  Select color
  
    {{ option }}
  

As mentioned above, you should use the compulated attribute for styles.
This will also automatically reflect any changes to the prop.
If you have certain conditions, you can also modify the values ​​based on the calculated values ​​in the callback function. I've added a Dark Mode example to illustrate this approach.

export default {
  data(){ return {
        selected_title_color1:'',
        titleColor1:'',
        colors:['紅色','藍色','黃色','粉色','土地色','水藍色','灰色','淺綠色','橙色'],
        colors_dic:{紅色:'red',藍色:'blue',黃色:'yellow',粉色:'pink',土地色:'blawn',水藍色:'aqua',灰色:'gray',淺綠色:'palegreen',橙色:'orange'},
        modalVisible1: true,
        darkmode: false, // example value
  }},

  watch:{
    selected_title_color1: function () {
      this.titleColor1 = this.colors_dic[this.selected_title_color1];
    }
  },

  computed: {
     // Here comes your style magic.
     // Return an Object that you will bind to the style prop.  
     // Changes to any reactive value will be reflected directly.  
     style(){
        // create the base style here.        
        let newStyle = {}

        // apply your titleColor1 which results in style="{color:titleColor1}"
        if(this.titleColor1){
          this.color = this.titleColor1
        }

        // if you would have more conditions you can add them here, too
        // just an _example_ if you would have a darkmode value in data.
        if(this.darkmode){
          this.backgroundColor = '#222222'
        }          ​
            ​
       ​return newStyle
    ​}
 ​},

 ​methods: {
    ​// rest of methods if 
 ​}
}

Then use :style="style" to append it to your div.

     ​

       ​
{{titleTxt1}}
       ​
{{contentTxt1}}
       ​
{{endTxt1}}
       ​{{buttonTxt1}}
     ​

Tips from my side. Instead of using an observer, I would outsource the code for setting the color and bind that method to the event that changes the color. This can make your application more flexible and clean it up. But what you do works too.