Trademarks Must Be Used To Remain Valid
Generally speaking, unlike a copyright, patent or trade secret, a trademark must be used in commerce in association with sales of a particular good or service to remain in force. So, one cannot have a trademark equivalent of a patent troll or copyright troll in the same way. If a trademark is not used in commerce for an extended period of time, it lapses.
This said, trademarks can be licensed and the use in commerce could be via a license, rather than through direct use. So, for example, HQ CORP might own the trademark, bring suits to enforce the trademark and might license the trademark to actual vendors of the licensed goods or services, while AFFILIATE CORP might actually sell the licensed goods or services and pay a royalty to HQ CORP.
In General, A Trademark Owner Is Not Required To Sue Everyone Equally
A company that owns a valid trademark certainly does not have to sue every single infringer, although failing to sue infringers who are legion, at some point, risks that possibility that the trademark will become a generic term and lose its protection. Subject to this limitation, however, a company that owns a valid trademark can develop any litigation strategy that it wishes to enforce its trademark rights.
Troll Style Litigation Models
A Troll's Business Model
The real concern in the situation you identify, is that it has a litigation strategy designed to enforce a trademark that is actually ineffective or void in the situations where suit is brought in order to extract settlements from defendants who can't afford to litigate their winning defenses to the lawsuits on the merits, in much the same manner that patent trolls often attempt to litigate. If someone attempts to mount a meritorious defense, they will typically drop the case or attempt to settle on very unfavorable but undisclosed terms.
The valid defense (usually known to the trademark holder at least after one or two cases have been litigated) could be that they are suing people with marks that don't actually infringe because they include only part of the protected mark that is generic, or it could be that the trademark is no longer valid because it is no longer being used in commerce (with the registration maintained, for example, based upon false declarations filed to keep the registration in force).
Defenses To The Troll Business Model
There is limited precedent for a business that as a business model seeks to bring meritless IP litigation for the purpose of securing settlements being subjects to consumer protection lawsuits (often under laws adopted in direct response to this business model) and for such entities being sued as corrupt criminal enterprises (in which the tricky part is determining precisely which conduct in the business model constitutes a predicate criminal act, often some kind of fraud). Such businesses are a subset of patent trolls when the IP in question involves patents.
There is also limited precedent for courts discerning the business model and mercilessly punishing the company bringing the suits and their lawyers for relatively minor technical procedural violations or for out and out bad faith with the resources available to courts to punish litigation misconduct.
An example of this in the U.S. is the Righthaven litigation in which newspapers licensed rights to sue for copyright violations related to their articles (ultimately unsuccessfully) to a copyright troll company.
Some of the better collective defenses to this approach are to share information about other litigation by the same firm, and then to cut and paste on a repeated basis defenses that have worked in other cases, particularly offering up to courts the fact that the company has lost almost identical cases in the past to give courts a roadmap to dismiss the claims in lieu of a settlement. This can make the affected group better off, but requires some collective cooperation and probably donative financial support to allow individual targets to be able to afford to effectively defend themselves to work.