Does a TR multi-entry visa to Thailand expire or renew with each entry?

No visa works that way. There is a difference between the validity of the visa and the duration you are granted when entering. A visa allows you to enter the country up to the expiration date. As long as you enter before then you will be stamped in for the duration that the visa allows. So in the case of a tourist visa you can enter on the last day and still be granted 60 days, which you can then extend for another 30 days by visiting an immigration office in thailand.

You need to use this visa before it expires in July! Once you've entered you get stamped in for a 60 day visit. It's multi entry so you can leave and re-enter for 60 days.

If it expires next month, does it mean you have already had it for 5 months?

The U.S. embassy does not issue visas for Thailand. You would have had to apply at a Thailand embassy for a visa or the DTV.

A multiple entry tourist visa is valid for 6 months from issue. You get 60 days on each entry. If you enter just before its expiry date you can milk almost 9 months from that visa.

The expiry date is fixed.