I made a trickster cleric with spy background. This caused an overlap of stealth and disguise. (Further compounded by me wanting to take hide in shadow ... which again granted me stealth.)
What happens when you are granted a skill from multiple sources?
The Player's Handbook and Basic Rules state you just take the skill of your choice in place of the duplicate proficiency (Basic Rules, p. 38; PHB, p. 126):
If a character would gain the same proficiency from two different sources, he or she can choose a different proficiency of the same kind (skill or tool) instead.
From the Basic Rules, page 38, under Proficiencies (in the section on Backgrounds):
Each background gives a character proficiency in two skills. Skills are described in chapter 7.
In addition, most backgrounds give a character proficiency with one or more tools. Tools and tool proficiencies are detailed in chapter 5.
If a character would gain the same proficiency from two different sources, he or she can choose a different proficiency of the same kind (skill or tool) instead.
So in this situation, where you receive proficiency with the disguise kit from 2 different sources, you simply choose another tool to gain proficiency in. Note that this has to be a tool proficiency, not a skill proficiency.
Backgrounds can be customized by the player, as noted on PHB p. 125 and Basic Rules p. 38:
You might want to tweak some of the features of a background so it better fits your character or the campaign setting. To customize a background, you can replace one feature with any other one, choose any two skills, and choose a total of two tool proficiencies or languages from the sample backgrounds. You can either use the equipment package from your background or spend coin on gear as described in the Equipment section. (If you spend coin, you can’t also take the equipment package suggested for your class.) Finally, choose two personality traits, one ideal, one bond, and one flaw. If you can’t find a feature that matches your desired background, work with your DM to create one.
You can customize the background by trading out the two skill proficiencies for any other two.
