Salary is sort of like market value of a home.

Year after year, the tax authority relies on surveys and general data about your neighborhood to estimate the fair market value of the property. On occasion, though, your house might get a more direct valuation. Maybe you got a formal appraisal. Maybe, and this is better still, you received an offer on the property. With this new, direct, specific data, you have better numbers than the appraisal board. If they don't match, you can take that data and get your appraisal changed.

Some companies may underpay and rely on inertia to keep staff. Others may genuinely think they are paying you a fair market rate, because they rely on salary surveys and other ancillary data that is about your role and general experience level but not about you. Your job offer from somewhere else is an offer letter to buy your work, and gives your existing employer new data with which they can evaluate you specifically in the job market. In this case, I don't think them "underpaying" you before was nefarious, and a counter offer could be made with complete sincerity.