You don't mention it but presumably you have acquired the domain name that you're going to trade through (my company.ie)?
The legal name of their company is of no consequence if they aren't actually trading under that name, which as best you can ascertain they aren't. The main legal issue that you might be exposed to, is of "passing off" - essentially, it's all about who actually trades using a name first and has a business that they can argue is being infringed by the latecomer's actions. Just setting up a company with a particular name is of very little relevance in that regard.
Here's what Google Gemini has to say about it!:
Trading Names
In Ireland, a trading name is a name used by a business that is different from its registered company name. Businesses may choose to use a trading name for marketing or branding purposes. While there is no legal requirement to register a trading name in Ireland, it is important to ensure that it does not infringe on existing trademarks or cause confusion with other businesses.
Passing Off
Passing off is a legal concept in Irish commercial law that protects the goodwill and reputation of a business. It occurs when one business misrepresents its goods or services as those of another, leading to consumer confusion and potential damage to the original business.
To succeed in a passing off claim, the plaintiff must typically prove three elements:
- Goodwill: The plaintiff must demonstrate that they have established a reputation or goodwill in their goods or services.
- Misrepresentation: The plaintiff must show that the defendant has made a misrepresentation, either intentionally or unintentionally, that is likely to deceive consumers.
- Damage: The plaintiff must prove that the misrepresentation has caused or is likely to cause damage to their business, such as loss of sales or reputation.
Passing off can occur in various ways, including:
- Using a similar trading name or logo
- Imitating product design or packaging
- Making false claims about the origin or quality of goods or services
Remedies
If a court finds that passing off has occurred, it can award various remedies to the plaintiff, such as:
- Injunctions to prevent the defendant from continuing the misrepresentation
- Damages to compensate the plaintiff for any losses suffered
- Account of profits to recover any profits made by the defendant from the misrepresentation
Important Note: This information is intended for general informational purposes only and should not be considered legal advice. If you have specific questions or concerns about trading names or passing off in Ireland, it is always best to consult with a qualified legal professional.