No, There is no legal requirement to Restrict Users in this way
There are several principles of trademark law which lead to this conclusion. Some of them would apply even if the "cards" mentioning trademarks were supplied with the app. These principles are common to trademark laws in many countries, specifically to the US and to most countries in Europe.
Use in Trade
Firstly, trademarks are only protected when used in trade or commerce. That is, when selling something, advertising it for sale, or promoting it for possible sale. "Sale" may include free distribution to the public. To a limited extent harming the reputation of a product benign offered for sale through use of its trademark may also be prohibited.
As I understand the description of the app in the question, none of the mentions of trademarks would be used to sell or promote the sale of the app or of anything else, nor to identify the app or anything except the trademarked product to which the mark properly applies. Thus such use is not infringement of any trademark.
Secondly, so-called nominative use of trademarks is permitted. This means using a trademark to identify the product it is properly associated with, when discussing or referring to that product. If one says "I drank a Coke with lunch." that is nominative use, and not an infringement of the mark. This also applies to competitive statements. The makers of a drink HyperCola, which competes with Coke, may advertise "HyperCola tastes better than Coke." and this is not infringement even if the makers of Coke do not agree.
The possible uses of a trademark described in the question would appear to be varieties of nominative use, and so are not trademark infringement.
Thirdly, for the use of a trademark to be infringement, there must generally be actual or potential consumer confusion. Reasonable people must believe, or be likely to believe, incorrectly, that the product or service with the unauthorized mark is the product normally associated with the mark (the "authorized product"), or comes from the same maker or source as the authorized product, or is endorsed or approved by the makers of the authorized product, or is in some way associated with it.
The trademark uses described in the question do not appear likely to create such confusion, and so would not be infringement.
No Duty to Police Users
Fourthly, the maker is not responsible for the independant acts of users.
The above principles would all apply even if the use of the unauthorized trademarks were being done directly by the maker of the app, and such trademarks were included in the app itself. But they are not.
As described in the question, any trademarks would be entered into the app by users of the app, and not suggested by the app or its maker. Even if such trademarks were used in an infringing way, the maker of the app would not be responsible for that, any more than the make of a word processor would be responsible if someone used that to draft plans to rob a bank, or to write a work that infringed soemnone's copyright. The maker of a software product is not required to police the users for possible unlawful use. Unless the product is marketed to suggest such unlawful use, or has no significant lawful and legitimate use, the maker is not liable for unlawful uses.
The maker of such an app has no legal need to restrict its users from entering trademarks as part of self-designed "cards". This is a good thing, because it would be very hard to reliably distinguish between legitimate and infringing uses of trademarks by users. Even keeping up with what terms are currently valid and active trademarks in every country where the app might be used would be a burden, as trademarks go in and out of use, and may be different in different jurisdictions.
That the app is free of charge is not really relevant to this conclusion. The same principles would apply if the app were being sold for money.