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Do items such as marked cards or loaded dice have any mechanical benefit? What is the purpose of these items if they do not provide such bonuses? Lastly, is there any advice in handling these items if there are no rules to support said items?

Additional Information:

For the record, I'm talking about in-game marked cards/loaded dice. Aforementioned items are listed in certain backgrounds, such as the charlatan background.

Do items such as marked cards or loaded dice have any mechanical benefit?

Not as written, no. The implication is that you should be able to get some kind of benefit when using them in an ability check, like when using the Gambling Rules in Xanathar's Guide to Everything, but what benefit this would constitute is not stated.

What is the purpose of these items if they do not provide such bonuses?

Many of the items provided by a character's background mostly serve as Role-Playing tools. The Soldier background provides items like an insignia of rank, which itself has no mechanical rules stated, but which most DMs often rule can provide Advantage in contextually appropriate situations, like trying to gain access to a restricted site or intimidating/persuading someone by appealing to authority.

Lastly, is there any advice in handling these items if there are no rules to support said items?

The easiest way to rule this is to simply provide Advantage on Ability Checks that make use of this item. So, for example, if you were using the Gambling Rules in Xanathar's Guide to Everything:

The character makes three checks: Wisdom (Insight), Charisma (Deception), and Charisma (Intimidation). If the character has proficiency with an appropriate gaming set, that tool proficiency can replace the relevant skill in any of the checks. The DC for each of the checks is 5 + 2d10; generate a separate DC for each one. Consult the Gambling Results table to see how the character did.

—Gambling, Xanathar's Guide to Everything, pg. 130

It's easy to rule that use of one of these types of tools would give Advantage to these rolls, perhaps with a particularly harsh consequence for failure, just as an example that doesn't make these rules especially complex. What you actually do ultimately is your prerogative as DM; If you want to evaluate the merits of a particular ruling, you're welcome to submit a question with your proposed ruling and solicit feedback, per the standards for Homebrew/Houserule questions

They are niche items for RP

Many items, like the Dragonchess set or the Three-Dragon Ante set have no mechanical explanation of what they do or how they work but are included in the game anyway.

They can be used in creative ways by the players as RP tools. Wizards of the Coast actually published a Three-Dragon Ante game with rules at one point, but the rules for D&D 5E don't specify how to play.

That's not to say that your table couldn't come up with a mechanical benefit to using these items, but since they do not seem to have any particular rules regarding their use, it's really up to you.