You said "Limited", which is functionally a synonym for restricted. An ability being limited to a subset of people is the same as it being restricted to that subset of people.
Regardless:
((Edit: Spoiler tagging the quote block since this is not really pertinent to the original topic of the thread at this stage))
Phb pg 125, "Bckgrounds", sub section "Proficiencies"
Each background gives a character proficiency in two skills (described in chapter 7, "Using Ability Scores").
In addition, most backgrounds give a character proficiency with one or more tools (detailed in chapter 5, "Equipment").
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.
Phb pg 125, "Backgrounds", sub section "Customising a background"
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 chapter 5. (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.
Phb pg 154, "Tools"
A tool helps you to do something you couldn't otherwise do, such as craft or repair an item, forge a document, or pick a lock. Your race, class, background, or feats give you proficiency with certain tools. Proficiency with a tool allows you to add your proficiency bonus to any ability check you make using that tool. Tool use is not tied to a single ability, since proficiency with a tool represents broader knowledge of its use. For example, the DM might ask you to make a Dexterity check to carve a fine detail with your woodcarver's tools, or a Strength check to make something out of particularly hard wood.
Phb pg 154, "Tools", sub section "Thieves' Tools"
This set of tools includes a small file, a set of lock picks, a small mirror mounted on a metal handle, a set of narrow-bladed scissors, and a pair of pliers. Proficiency with these tools lets you add your proficiency bonus to any ability checks you make to disarm traps or open locks.
Phb pg 143, "Starting Equipment"
When you create your character, you receive equipment based on a combination of your class and background. Alternatively, you can start with a number of gold pieces based on your class and spend them on items from the lists in this chapter. See the Starting Wealth by Class table to determine how much gold you have to spend.
[Tables and lists include 'Tools', Thieves' Tools cost 25g]
The Phb tells you, clearly and openly throughout the course of character creation, that anyone, regardless of their race or class can have proficiency and access to any set of tools or skills that suites their character, including thieves' tools... one class gets them from a class bonus, but that is only one source, and it is in no way, at any point, ever implied to be limited to them in any way. I would very much appreciate it if you would acknowledge that, and that your previous statement - when you said you wanted it limited, and someone asked you if it was a thing in 5e, and you told them that yes, it was, unless the dm homebrewed, was a (perhaps unintentional) mislead.
