Updated: Apr 24
The short and precise answer is "no" - they have different names, use different alphabets and we could just leave it there, but that would make for one terribly short blog post and rob you of some of the fun facts and curiosities we'll try to capture here below.
Rest assured that this is just a snippet into the wonderful world of words and to properly understand the relation between the ways people speak you'll need to do more serious research on your own.
Origins of the Latvian language
First and foremost we have to talk about the origins - both Latvian and Russian belong to the Indo-European language group, so some similarities are inevitable, just like between Latvian and Swedish or Latvian and Portuguese for that matter. Going deeper in we'll see a language group called "Balto-Slavic", here linguists and historians (and everyone in between) are split - no one can tell whether Baltic and Slavic languages where the same "long time ago", then split apart and as a result have retained some similarities or they were never part of the same sub-group and just geographically met "not that long ago" and as a result borrowed or loaned some words, which, as we see in today's globalised world, is kind of inescapable even if you are not neighbours.
Be that as it may, once you narrow down even more you'll be left with the Slavic languages (Eastern, Western and Southern) and the Baltic group. Today there are only two actively used Baltic languages - Latvian and Lithuanian, both representing East Baltic languages while the West Baltic languages like Old Prussian or Sudovian have past into extinction over 200 years ago, largely due to the prevalent German infliuence in their lived areas (modern day Northern Poland, Kaliningrad and Lithuanian west coast), so today we only have texts and no audio records to reconstruct the original pronunciation of these long gone tongues.
Although Latvian and Lithuanian are related, they are not mutually intelligible, so in daily communication with our neighbours we'll use either Russian or English (mostly depending on the age of the speaker). Things change a bit once you are nearer the common boarder - just like all over Europe and beyond, those who live in a close proximity to another country will learn just enough to make their shopping trips "cheaper" and their dating options more "eclectic".
So now that we know of the common linguistic origins and how even the related Baltic languages do you no good to understand Latvian, one could ask: are there any similarities between Latvian and Russian at all? And the answer is pretty much the same if you'd ask about the similarities with Polish, Swedish or German vocabularies - yes, there is plenty in common, but you might not pick it up at the Central Market bustle or recognise same-origin words while browsing a book due to the borrowed lexicon being pronounced and written in a "Latvian way". That means things like adding "s" at the end of masculine nouns, "softening" certain "harsh-sounding" parts, putting emphasis on the first syllable (a trait which Latvians themselves borrowed from the Finno-Ugric Livonians) and so on.
Similarities between Latvian and Russian language.
Probably the most similarities between Russian and Latvian will be found in what's the easiest to observe - words describing people and their body parts, nature and the processes that tie it all together. Russian "человек"(chelovek) becomes Latvian "cilvēks" (cil-vehx ?), "голова" (golova) - "galva", "ветер" (veter) - "vējš" (vehy-sh). Counting from 1-10 is nearly identical (most likely to not be cheated by rogue foreign merchants from the East over a millennia ago) and so on. These words have been in use for centuries and have blended in so well noone would even notice, if not for this blog post ;)
A very different case are the relatively recent Russian influence that Latvians have acquired during the Soviet era: here quite often the loan-word (or barbarism, if you're really making a case against this practice) in question has retained its full original pronunciation and as a result doesn't "sit well" with the rest of the sentence. Sometimes that is the goal, as Latvians often use Russian swearwords due to not having any of their own (try and offend a Latvian in Latvian, they'll probably just laugh you off). More commonly we're talking about words that "break up" longer sentences or try to bring attention to whats about to be said next: instead of using the Latvian "Lūk" (pronounced "Luke", just like the Skywalker) which means "Here", many Latvians stubbornly will employ the Russian "Вот" (Vot), right before making their big point.
There are of course many other angles one could look at this subject from: syntax and how it sometimes migrates from Russian into Latvian due to a poor translation, borrowed words that only found home at certain region and so on. Let's leave that for another day and for now conclude: Latvian and Russian are definitely not the same, however, just because the resemblance isn't obvious, doesn't mean it's not there at all.