There’s tense as in a verb form and tense as in OMG-I-just-sent-out-a-full-MS-to-an-agent-and-now-I’m-going-crazy. They don’t seem to have much to do with each other, do they?
The latter tense first showed up in its adjective form in the 1660s, at first meaning physical tension, two centuries later also meaning emotional distress. It came from the classical Latin tensus, the past participle of tendere or “to stretch.” As for the verb form, it showed up in the 1670s, about a decade after tense the adjective came out. Because everyone likes verbing words.
What about the other tense, the one related to parts of speech? Could they be related.
No. What are you, crazy? That might make sense. This is etymology we’re talking about.
The other tense showed up much earlier, in the early fourteenth century, and it comes from the Old French tens, which means time. Tens can be traced back to the classical Latin tempus, which should surprise no one who’s heard the phrase tempus fugit is also the origin word for temporal.
TL;DR: they sound the same because the Old French said “tens” instead of “tems” when referring to verb tenses.