It depends. You need to understand some basic legal points. Registration for a "DBA" does not mean that you have the power to exclude someone else from using the name. Most states require registration of a "DBA" for purposes of assuring that members of the public can locate the business owners for the purposes of, inter alia, filing legal claims against them. However, registration of a "DBA" does not mean that you are only entity in your city, state or the country entitled to use the name.
The issue here arises not because of your DBA, but because of potential claims that you might have under trademark law. If your name has become associated with your company's goods and services within a specific geographic area, and you are the first company to have begun using the name in this geographic area, then you may have become entitled to common law trademark protection for your name. Common law trademark protection arises from use of your name in commerce, and you become entitled to it when the name (or logo with a name) that you use becomes associated with your company's goods and services in the minds of consumers. Of course, it would have been better if you filed Federal trademark registration, because that give rise to a presumption that you are entitled to use the tradename nationwide for the classes of goods indicated in your registration. But common law trademark rights can sometimes be asserted successfully even without federal registration.
However, you can only assert common law trademark rights against this other company if (a) you began to use the name in commerce before the other company, and (b) you can show that the other company's use of the name has created a likelihood of consumer confusion. If the other company began using the name before you and is the senior user, it would be almost impossible for you successfully assert that it should stop using the name (by the way---the date of filing the DBA registration is an insignificant factor in this analysis---what matters is when actual use began in each business). Secondly, if the two companies do not compete in the same geographic area, if they do not compete for the same customers, and if there is no real prospect of consumer confusion, then you probably will not be successful in asserting that it should stop using the name.
In most cases such as this, the critical issue is whether there is consumer confusion---are you losing customers because the other company uses this name? If you are not competing in the same geographic vicinity, if you are not providing the same services or selling the same products, or if you are not competing for the same customers, then there is a low probability of consumer confusion and the two companies can probably continue to use the name simultaneously.
These issues are complex and sophisticated. Before giving definitive advice on this question, any lawyer would need much more information from you. The general advice that you receive on this web-site is no substiitute for the advice that you receive when you retain counsel to represent you. In this situation, it certainly is in your best interest to retain counsel to represent you in this matter. Don't forget--the other company may also be upset at your use of the name. Unless this is handled carefully, this could result in an expensive law suit. The investment you make in retaining counsel will likely save you a lot of money in the long run.
If the two companies do not compete with each other and if their actual and potential customers are not likely to be confused by their uses of the name, then it might be ok for the two companies to use the same name.
On the other hand, if the name has become a trademark associated your company, if you started using the name before the other company, and if the other company's use of the trademark creates consumer confusion
You may be mixing apples and oranges. Business names aren't necessarily company names which aren't necessarily trademarks. There are several ways to "register" a company name.
You have a DBA. That's a fictitious business name for you to do business under. Your County recorder may not allow more than 1 person or company to use the same DBA because then consumers won't know which DBA is which when they look them up.
This other company may be a corporation or an LLC, and the state Secretary of State won't allow more than 1 person or company to us the same nusiness name for the same reason.
Then there are trademarks, which can be registered with the state, or with the federal US Patent and Trademark Office. Both you and this other business may be using the same TM, although it seems that you're providing a service, and would have a service mark, and this competitor is selling equipment, under a trademark. Because you're both in the landscaping business, I think a tradenark examiner reviewing the TM applications would find CONSUMER CONFUSION to be likely, and that's the essence of trademark infringement.
Whether you have superior TM right to this other company, depends on who was in business first, and whether you used this name as TM or not in interstate commerce (maybe in an online website, even if your services were rendered only in TX) or in your own state's commerce.
You'll need to see your own TM lawyer for specific advice.
PLEASE READ THIS BEFORE YOU COMMENT, EMAIL ME OR PHONE ME. I'm only licensed in CA. This answer doesn't make me your lawyer, and neither do follow-up comments and/or emails and/or phone calls --- we need an actual agreement confirmed in writing before any attorney-client relationship is formed. Please don't expect me to respond to your follow-up queries. This answer doesn't constitute legal advice, and shouldn't be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue.