Yes, tendering is giving. But it's usually used more in the sense of giving or offering something formally after thoughtful consideration and in expectation or hope of acceptance. I probably wouldn't say that I tendered you a book, flowers, or money. I mostly hear it in the context of submitting a letter of resignation from a job or position. "He tendered his resignation" means he submitted a letter of resignation and awaited (or is awaiting) a decision on the part of his employer. Something tendered might be accepted or rejected.

It's more poetic than everyday conversational, but I could tender you my thoughts (or anything else). That would imply that I formally offer them to you for your consideration and acceptance.

I'm not sure why someone would say "tender" rather than "give" a vote, other than to add the element of formality and offering. In my part of the world we seldom say or hear either "tendered a vote" or "gave a vote", but just "voted".

http://fluentli.com/questions/tender-a-vote-vs-give-a-vote